Trin Coll. ^<1>
Wednesday evening; Nov^r 11^th

My dear Father,

I am much obliged for the other copy of the paper which arrived here, the other day. I have several things to show you, but it takes too much time to write them but as they are in general all written down they wont be forgotten. Gillespie showed me, in the case of the p theorem, of that the points in which the inscribed & excribed circles touch the base of a ∆ [illegible deletion] are the vertices of an a hyperbola & th an ellipse confocal to it, that all th it is proved readily for the p^ts of contact of the 3 excribed circles in exactly the same manner as you I proved it for the inscribed circle which is therefore I think a better proof than the one we had. I think I shall have my 3 days subjects up pretty well. I have tonight been reading the beginning of Sir David Brewsters life of Newton ^<2> & find it very interesting.

I will mention one of my theorems deduced, from those we considered. If any number of parabolæ p having the same focus and their axes in the same line intersect they cut at right angles. If 2 turned one way cut 2 others turned the others turned the other way, the diagonals of the curvilinear quadrilateral are equal. also if one diagonal passes through the focus the other is parallel to the axis. If two ellipses have one of their foci only coincident, they may be said to be confocal.

So supposing those you investigated, were called con’-centro-/con-focal [sic] or con-foco-con-centric, how would you distinguish that class of e ellipses which have one focus & its directrix coincident and different excentricities. They might perhaps be found to possess some curious properties. I have been able to apply analysis to coni investigating conic sections more than I could before, and differential calculus to a certain extent – but I obtained no very pleasing result when I tried to get the radius of curvature of particular curves by the differential calculus, though I had the correct formula in Todhunter. ^<3> It would come out an unmanageable qu expression. I cannot certainly expect to be high in the Tripos ^<4> but I hope at any rate not to be gulphed. I should get on better if I were a better hand at getting up in the morning which is a process that I do not understand. however I find a course of dumb-bells is a good method of awakening oneself. Will you please thank Ela ^<5> for her last letter and also one I recei Monnie ^<6> [sic] for one I received some while ago and I do not think I acknowledged. I hope Mamie is better than she was by the last accounts I heard. If you could spare Salmons Conic Sections, ^<7> a little later on it might be useful to me. It would not I think be easy to borrow a copy here and it is desirable to have the last edition. I think I might get something out of it. and If if I could get a a knowledge of porjections projections it might help me to see through some problems riders, and if desirable afterwards prove them by geometry.

Here is a thing which seems to be true but I do cant prove it. it always comes right the same when drawn. [illustration]

If ABCD be a quadrilateral circumscribing a [illegible deletion]

EFGH being points of contact.

Is it truee true that if the opposite angles AC. & BD be joined and also EG. & HF. that then these lines all pass through the same point. O. It seems to be true. Can it be proved by projections?

There is some great dodge of projecting the pts of intersection s of the opposite sides to infinity but I dont understand it. We have to go on and th do papers very frequently some of which I have done tolerably; others not. The Students guide is a very useful book in showing one what one had better do.

Your affect son
Charles.

I think it possible that I might get a place in the Classical Tripos, there being about 6 weeks (I think) to cram in, but six weeks is not much; and I am not certain that I care to undergo that process of severe cramming. Certainly though I have read read a certain fair amount of Classics yet they are not up. Moreover the it is not the idea of these Tripos Tripos’s that men are to get into them by cramming up the subject a short time before when they havent kept it up. So I am inclined to think I will settle to drop it. What do you think?
P.S. Thursday morning.
Now there is one matter of considerable importance, to be decided. Whether I shall try and get a place in Classics after the mathematical Tripos is over. At first I thought I would, but on the whole perhaps I am inclined to think that I had better not. The point of the matter is that this must be decided at once, and I should be very glad if you can, if you will let me know what you think by return of post, for M^r Mathison ^<8> has sent to know when I mean to give up my rooms; and of course I can and that requires to be answered at once.

[envelope:]
H Fox Talbot Esq
Millburn Tower
nr. Edinburgh

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Notes:

1. Trinity College, Cambridge.

2. Sir David Brewster (1781–1868), Scottish scientist & journalist, The Life of Sir Isaac Newton (London: William Tegg, 1861).

3. Isaac Todhunter (1820–1884), mathematician, Treatise on the Differential Calculus and the Elements of the Integral Calculus (1852).

4. Cambridge honours examination.

5. Ela Theresa Talbot (1835–1893), WHFT’s 1^st daughter.

6. Rosamond Constance ‘Monie’ Talbot (1837–1906), artist & WHFT’s 2^nd daughter.

7. George Salmon (1819–1904), mathematician. [See Doc. No: 08067].

8. William Collings Mathison, tutor at Cambridge.