(r 0 2 — p 0 2 ) P — —p 0 (pp' — qq' + rr' — ss') + r 0 {pr + p'r — qs' — q's), 
» R= r 0 ( „ )~Po( „ ), 
„ Q= Poipq' - p'q + rs' -r's)-r 0 ( „ ), 
» S = -r 0 ( „ )+Po( » ). 
On writing in the equations u' = 0, then P, Q, R, S, p', q', r', s' become = p, q, r, s, 
Po, 0, r 0 , 0; and the equations are (as they should be) true identically. The equations 
may be written 
u+u' u—v! u u' u u' u u' u u' u v! u v! u v! u u' 
(c 4 c 4 ) ^ ^ c 2 (^ 2 .^ 2 ^ 2 .^ 2 ^ 2 .^- 2 ^ 2 .^ 2 ) c 2 (^ 2 .^ 2 ^ 2 .^ 2 ^ 2 .^ 2 ^ 2 .^ 2 ) 
(8-4) 4 4 = -4(4.4-7.7 + 8.8 -11.11) +8(4.8 + 8.4-7.11-11.7), 
( „ ) 8 8 = +8( „ ) — 4( „ ), 
( „ ) 7 7 = +4(4.7-7.4 + 8.11-11.8) -8(4.11-11.4 + 8.7 - 7.8), 
( „ ) 11 11 = — 8 ( „ ) + 4 ( „ )• 
There is of course such a system for each of the 60 Gopel tetrads. 
Differential relations connecting the tlieta-functions with the quotient-functions. 
134. Imagine p, q, r, s, &c., changed into x 2 } y‘\ z 2 , w 2 , &c.; that is, let x, y, z, w 
represent the theta-functions 4, 7, 8, 11 of u, v; and similarly x', y', z', w' those of 
u', v', and x 0 , 0, Zq, 0 those of 0, 0. Let u', v be each of them indefinitely small; 
and take d, = u' ^ +v , as the symbol of total differentiation in regard to u, v, 
the infinitesimals u' and v' being arbitrary: then, as far as the second order, we have 
in general 
S- (u + u', v + v') = ^ (u, v) + 8^ (u, v) + l-S 2 ^- (u, v), 
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