M1M1 Treasure Hunt

The M1M1 treasure hunt, week 9

Chapter 6: The men from C.I.Ba.

"Last week's escapade was horrible," complained Ann to the Doctor. "If I'd wanted to meet metal monsters, I'd have read Mechanical Engineering."
"But why did you come to Imperial at all? A very high proportion of attempted alien invasions of the Earth seem to take place in London. It's a statistical anomaly. Hey, your name, Ann O'Malley! I've only just −"
"Yes, frightfully amusing, isn't it. And believe it or not, avoiding alien invasions wasn't high on my list of concerns when completing UCAS. Could we go somewhere quieter?"
"OK. How about Switzerland? Very peaceful place. Timeless mountains, never-ending waterfalls...nothing changes. Not until the 1970s did they get round to allowing women to vote, and not wishing to rush things, they joined the UN in 2002. Time seems to pass more slowly there. Maybe that's why Einstein felt so at home."
"You promise? No alien monsters?
"You have my solemn word there are no alien monsters in Switzerland at this time. Let's go visit Uncle Albert, in 1904, say."
"Uncle Albert?"
"Well, he said everyone's a relative, didn't he?" chuckled the Doctor, and sent the blue box on its journey.

Ann opened the door and beheld a young but easily recognisable figure waiting impatiently by the side of the road near a large clock.
"Typisch! Das Tram ist schon wieder zu spät," he complained.
"At school they taught us to say die Straßenbahn, or sometimes die Tram," whispered Ann to the Doctor. "That must be a local dial... − I mean variant," she hastily corrected herself, glancing around nervously. The automatic translation then started as Albert continued,
"I want to do a thought experiment with that clock, before I have to get back to work."
"Why do you need a real tram for a thought experiment?" began Ann, but just then the tram appeared. The three of them got on, and Albert began to explain:

"You see that clock? Light from it travels like a sinusoidal wave. When it reaches our window some of it goes through, reaching us as eix but some of it is reflected. The reflected wave bounces back off the shiny clock face. Then some more of it reaches us, adding to our original wave, but this time weaker and with a different phase, which we can treat as Rei(x+α). Then it keeps bouncing, next time it reaches us as R2ei(x+2α), and so on. So the total amount of light which comes through is an infinite series. You follow me?"
"Yes I think so, but you can just sum that series can't you?"
"Of course, and it sums to ϱeix, say. And if I'm looking through a window, when R is fairly small, that's ok. But what if most of the light is reflected so that R is very nearly one? Then ϱ can be very big, for example, as R → 1, |ϱ| → 1 when
α = ." ?

"But isn't it bizarre to look through an almost opaque window?" asked the Doctor reverentially.
"Yes, maybe. But I'm interested in photo-eclectic effects."

"But I must get back to the patent office," said Albert, glancing at his watch. "I've some visitors from Chemische Industrie Basel to deal with. I'd be glad to talk more when they've gone."
"The C.I.Ba. men?" exclaimed the Doctor worriedly. "What do they want?"
"Oh, they've invented some special ingredient they want to include in Swiss chocolate, which, as you know, is the best in the world. But before they can do this, it needs a proper patent. Which is where I come in. It used to take forever to register these patents, but I've made the whole process more efficient."

Ann and Dr Hu followed Albert back to the patent office, where the CIBamen were already waiting. They wore metallic white uniforms with matching fearsome facemasks.
Ann froze in terror. "Doctor! You promised...!" she hissed.
"Ah, you may may have misunderstood me slightly. I said there were no alien monsters in Switzerland at that time, that being then. The CIBa men are indeed fearsome aliens, around today but they were eventually defeated in the Ciba-Geigy wars of 1971. Whatever this chocolate business is about, it won't be good for humanity - we must stop them, or at least delay their plans."

"Now," began Albert to the chief Cibaman. "You are filing a patent for a liquid paste to be applied to the surface of chocolate bars. I gather it is chocolate-coloured, and contains bubbles of an inert gas, Neon. What are you calling it?"
"Brown Neon lotion," replied the Cibaman tonelessly.

"That does sounds interesing," Albert pondered, with a faraway look. "Hmm. I'll think about that later. But anyway, you also want to introduce a special flavour-enhancing compound X into the tiny bubbles. I'm sure that'll be fine, I'll just need to run the text past our legal department, to check it won't engender difficulties."
"Engender difficulties − that gives me an idea. You distract them, while I look at that document," insisted Ann.
"I, er, hello? That does look like a complex problem," began Dr Hu. "Maybe you can help me with one of my own. I've got this function, you see,
f(x) = sqrt(13 + 5 cos x) and I expand it as a power series in x but it doesn't always converge."
"Foolish human! |x| may not exceed R, the radius of convergence of the series," explained the cibaleader.
"Oh and what is its radius of convergence exactly? Somewhere about 1.6 I think," continued the Doctor, as Ann leafed through the patent application.
R = ?

While Albert and the Cibamen were distracted, Ann had made occasional alterations to the patent application. Soon, the cibamen marched out, satisfied that their diabolical master-plan was on course.

"What did you do?" murmured the Doctor.
"It's so difficult getting the gender right in German - even the Bernese Swiss and the Germans can't agree whether it's die Schokolade or der Schokolade − they are still arguing about it in 2014. So I've made the genders inconsistent in the patent. Neither the Germans nor the Swiss will accept the document as legal as it now stands and I don't imagine the lawyers will sort it out for a century or so."
"Brilliant! Trapped by lawyers! I almost feel sorry for the Cibamen. I wondered why they were so inactive in the 20th century."

Meanwhile, Albert had been reading the patent. "Interesting, they say the amount of energy, E, derived from eating chocolate is proportional to the mass, m, of Riboflavin, or vitamin B2, it contains, so that in suitable units

E = m(B2).

Not sure that's quite right, but mass and energy, hmm..."
"Can I tell him? Then it'd be me who discovered it!" breathed Ann, excitedly.
"No, you musn't meddle - I have reason to believe you'll discover something in your own right one day, but if I told you what it was it would change your future and you might not do it. Anyway, you probably think the formula is E = mc^2, but in fact
E = mc3/sqrt(c2-v2), where v is your speed, and c is very large.
So as I take us home you can work out that actually, as c/v → ∞,

E = mc2 + ." ?