1901 - 2 .] Dr Muir on the Theory of Orthogonants. 
253 
/ 
0P 
V(G + G'cos 2 P + G"sin 2 P) * 
As in the preceding paper, Jacobi does not begin with the 
substitution which is really sought, but with the reverse sub- 
stitution, — that is to say, the substitution necessary for the trans- 
formation of 
// 
sin P-0P-8# 
G + G' cos 2 P + G" sin 2 P cos 2 # + G'" sin 2 P sin 2 <9 
into 
// 
sin if/‘dif/-dcf> 
P ’ 
— knowing that from the latter substitution, when found, the 
former will be obtainable. This substitution he takes in the form 
cos P 
sin P cos 0 
a + a cos if/ + a sin ifr cos + a" sin if/ sin cf> 
8+8' cos if/ + 8" sin ifr cos </> + 8"' sin if/ sin </> ’ 
P + p' cos if/ + /3" sin i/r cos <£ + /3"' sin if/ sin 0 
8 + 8' cos if/ 4- 8" sin if/ cos </> + 8"' sin i//- sin <£ ’ 
sin P sin 0 = Z±X G0S ^ + y" sin if/ cos + y"' sin if/ sin 
8 + 8' cos i ft + 8" sin if/ cos </> + 8"' sin if/ sin </> ’ 
the three new facients, cos P , sin P cos 6 , sin P sin 6 being ex- 
pressible as fractions whose numerators and common denominator 
are linear functions of the original facients. It rests with him 
therefore to prove that the sixteen quantities 
a, a , a , a 
p, p, p, r 
r rr /// 
7 > 7 * 7 » 7 
8 , 8 ', 8 ", 8 '" 
und the four 
G, G', G", G'" 
are so determinable that the performance of the substitution may 
bring back the original integral. 
By reason of the fact that 
cos 2 P + sin 2 P cos 2 0 + sin 2 P sin 2 # = 1 
for all values of P and 0 , it follows that the expression 
(a + a cos if/ + a" sin ifr cos <f> + a” sin ifr sin <£) 2 
+ ({$ + P' cos if/ + p" sin if/ cos cf> + p'" sin if/ sin <£) 2 
+ (y + y cos if/ + y" sin if/ cos cf> + y" sin if/ sin <£) 2 
- (8 + 8 ' cos if/ + 8 " sin if/ cos <f> + 8 "' suiif/sm<f >) 2 
