Full text: The collected mathematical papers of Arthur Cayley, Sc.D., F.R.S., sadlerian professor of pure mathematics in the University of Cambridge (Vol. 1)

79] 
AND ON ANOTHER ALGEBRAICAL SYSTEM. 
469 
the result is easily shown to be 
l : m : 9 = A (B + H + 57 x ) : B (A + H + zr 2 ) : (H + OTj) (H + tz- 2 ) — AB. 
But the problem may be considered as the problem for two variables, analogous 
to that of determining the conic having a double contact with a given conic, and 
touching three conics each of them having a double contact with the given conic; 
and in this point of view I was led to the following solution. If we assume 
Bl — Hm = u, — HI + Am = v, 
Hence, writing K6 2 + ^ V = — s 2 , we have 
u = K6 + OTjS, v = KO + -3T 2 s, V + K 2 9 2 + Ks 2 = 0 ; 
and substituting these values of u, v in the last equation, 
A (KO + otjs) 2 + 2H(KO + Otis) (KO + '57 2 s) + B (K9 + vr 2 s) 2 + K 2 0 2 4- Ks 2 = 0, 
or reducing, 
K 2 0 2 (A + 2H + B +1) + 2K0s {(H + H) + (H + B) ot 2 } + s 2 (Anr-? + + Bvxg + K) = 0 ; 
whence 
[K6 (A + 2H + B +1) + s {(A + jH) + (H + B) S7 2 ]] 2 
= s 2 [{(yl + H) OTj + (H + B) ot 2 } 2 — (A + 2H + B +1) (Hw! 2 + 2ihr 1 sr 2 + B^ 2 2 + K)\ 
= S 2 {— K (vr 1 — '57 2 ) 2 — (A 57J 2 + 2 Htx x tb 2 + I?ct 2 2 ) — K (A + 2 H + B+ 1)}, 
= s 2 {— (A + K) 57j 2 — (B + K) Ts 2 — K (A + 2H + B + l) + 2 (K — H) vtivt»}, 
= s 2 {2'57 1 2 -5t 2 2 — K (A + 2H + B + 1) + 2 (K — H) ■5r 1 Br 2 }, 
= s 2 (vti'gtz + K — H) 2 :
	        
Waiting...

Note to user

Dear user,

In response to current developments in the web technology used by the Goobi viewer, the software no longer supports your browser.

Please use one of the following browsers to display this page correctly.

Thank you.