— 107 — 

 Cayley an Schläfli. 



Dear Sir 



I have to thank you very much for your letter of the 

 20"" Aug* , with the accompanying postoffice order for 82 Frs. 

 (fc the several luemoh-s which you kindly sent me. I am very 

 sorry the dispatch of the Journal has been so unfortunate. 

 I send herewith the missing N°^ 30, 31 & 39. I would have 

 written before, but I was away from Cambridge, among our small- 

 scale but very injoyable mountains in Westmoreland & Cumber- 

 land, & you mentioned that you were also away from Bern. I trust 

 that you have had an equally pleasant summer excursion. 



As tho the equation ü = AP -(- BQ , admitting that it 

 exist, the determination of the functions A, B could be effected 

 by indeterminate Coefficients: viz, if deg. £2 <^ deg. P -f- ^eg. Q, 

 then A & B will be completely determinate functions so that 

 the coefficients will be completely determined l)y the method 

 in question : but if deg. f2> deg.P-|- deg. Q, then any particular values 

 being A', B', the general values will be A' -j- K Q, B' — KP, 

 K Ijeing au arbitrary (integral) function of the degree ^= 

 deg. L2— deg. P — deg. Q. But I understand your difficulty is 

 that we have no proof of the existence of the equation £2 = 

 A P -}- B Q. I am not able to give one — and I can only 

 say that the existence of this equation with integral or frac- 

 tional values of A, B appears to meel axiomalic — Suppose 

 A & B originally fractional, then the form really is CQ = 

 A P -|- ß Q — l>Lit then P ^= o, Q = o does not of necessity 

 imply 12 = o, but only il = o or eise C ^= o ; and the System 

 (P = o, Q = o) breaks up into the two Systems (P ^ o, 

 Q :^^ 0, .Q = o) (P =: o, Q = o, C = o) contrary to the ori- 

 ginal supposition that P = o , Q ^= o is an indecompo- 

 sable System. A difficulty is that we might have C = .ß, viz. 

 PJ = A P -|- B Q, without n = A P -f- B Q - all this for what 

 it may be worth & I should be very glad if you can add anything 

 to the analytical proof. 



I have studied the Fourier-'nevieü question so little that I 

 do not venture any remarks upon the latter part of your letter. 

 I remain, dear Sir. your very sincerely A. Cayley. 



Cambridge, 11. Sept. 1871. 



