214 
ON SOME ANALYTICAL FORMULAS, AND THEIR APPLICATION 
[32 
write & = BG -F°~, (4), 
23 = CA -G\ 
<&=AB-H\ 
fi = GH-AF, 
(fit = HF - BG, 
=FG ~CH. 
<® = a, 23, on, §, p* (5), 
w&)/ = vK' — v'£> — K’%> Zv'— Z'v (6), 
we have 
W (m 1, tw 2 , Q)lT(c03, co 4 , Q) - Tf (&>!, &) 3 , Q)fF(« 2 , co 4 , Q)= If (&)i&) 4 , &) 2 a?3, (&) ... (7); 
of which we may notice also the particular cases 
W(wi, w 2 , Q) W (g> 3 , (o 3) Q) - IT(t»i, «3, Q)F(<w 2 , 0) 3 , Q)=W(w ] w 3 , 03^3, <©) ... (8), 
W (a> u 031, Q)W(o3 2 , 03 2 , Q)-{W(o3 u 03 2 , Q)Y= W 03,03,, (S) ... (9). 
To these we may join the following formulae, for the transformation of the function W. 
Suppose 
oh = aa?i + a'y 1 + a!’z x , bx 1 + b'y 1 + b'%, ex, + c'y x + c"z x (10), 
03, = ax, + ¡x'y, + a!'z 2 , bx 2 + b'y 2 + b"z 2 , cx, + c'y, -f c"z 2 , 
then, writing g =a , b , c (11), 
g'= a', b', c', 
g" = a", b", c", 
Pi = Xi, Vi, Zi (12), 
p 2 ~ x 2 , y 2 , z 2 , 
®=V(g,g,Q), W (g r , g', Q), W(g",g",Q), W(g',g",Q), W(g",g,Q), W(g,g',Q)( 13). 
we have 
W (o)!, 03 2 , Q)=W (p x> p 2 , ©) (14). 
Similarly, writing 
V = W (Yg", Yg", <&), w (g''g, Yg, (®), W (gg\ gtf, $2) 
W (W* S’ <&)’ W (Y£_> Wj Q)> w (7Y> S’ <&)> (15), 
we have W (03^, 03 2 03 3 , @)= W (pgh, p 2 p 3 , T 1 ) (16), 
in which equations may obviously be changed into Q.