On Systems of Rays. 41 
respect to o’, 7, v, and denote by 87},, the corresponding value of 87, determined 
by the coefficients (Z*), we may generalise these values by means of the following 
relations, analogous to (S") ; 
87. 1 =8T—80.y.T 80.9 7; 
P71 = 8 T—80.7, T-80'.—) T 
(L*) 
— 280.8y, T— 280'.87) T 
+ 60?.Wi(vi +1) T+ 202.00! qiyi T+ 802.91 (yi +1) 7: 
Vv Vis being here characteristics of operation, defined by the following symbolic 
equations, 
) ) } 
Rt Se, ta en Yeh 15 
(M*) 
— ks 1, o /o 1 
eC ae pat a 
More generally, if we denote by 7,,,,, the function deduced from 7 by the homo- 
geneous preparation mentioned in the sixth number, which coincides with 7" when the 
variables ¢ tv o' rv’ x are connected by the relations Q=0, Q’=0, and which is, for 
arbitrary valnes of those variables, homogeneous of the dimension 7 with respect to 
o, r, v, and of the dimension 7’ with respect to o’, 7’, v’, we have the following expres- 
sions, analogous to (U*), 
Of, §=89 1 —80.y, 1 — 0.9L; 
2 T= 8 T— 80.9, T— 80.9 T — B2.89n T — 280.89’, T (N’*) 
+ 607.Yn (Vn +1) T+ 280.82'.Ya Vn! L822. 7 WT wt +1) T: 
defining the characteristics V7, V’,', as follows, 
3 3 ah nk 
Va=oy tre tus ms Vu=esstr sts an. (0°) 
Reciprocally to deduce the coefficients of JV, of the second order, from 
those of 7, on the same suppositions of homogeneity, and with the same dimensions 
n=1, n =1, we are to eliminate 80, 6’, Sv, between the differentials of (G’) and 
(Z"*), and we find the following system, 
VOL. XVII. M 
