194 



also let 2 (m rfa), &c, be denoted by r x \, and similar expressions for 

 the others. 



Also the latter part will give 



" ~i v " 2m ~ y z ^> ( x2x * + y 2 y* + + 2x y x ^ 1 + 2xz x & + 2 y z y^) 



Multiplying these two factors together, and retaining all such terms as 

 are either of one dimension x and y, or of three dimensions, which are 

 the only ones which will produce terms of the form required, we 

 have 



" ~I IF r [f yi XlH + ^ 3 ~ 2z2 ^ + z * y % * + ~ xz ^ XlVlZl 



Now, 



x = a cos 0 - 0 + ft cos 0 + 0, and y = - a sin 0 - 0 - /3 sin 0 + 



^ 2 = 2 ( a2 +I/ 32 ) + 2 a? cos 20 - 20 + a/3 cos 2<p + a/3 cos 20 



+ i/3 8 cos 20 + 20 



Multiplying this by the value of y, and retaining terms whose argu- 

 ment is 0 - 0, 



^y = (" iO 3 + "P) + i a3 * i a P 2 ~ i a F) sin 0"^ = - i ( a3 + 2 /3 2 a) sin 0^0 



which is the coefficient of x?z x above. 

 Also in like manner 



f = - « (a 3 + 2/3 2 a) sin <p - 0 



and 



sin 2 / 



= — (1 - cos 20) (a sin 0 - 0 + /3 sin 0 + 0) 



2i 



= - I sin 2 * (2a - /3) sin 0-0. 

 Thus the coefficient of y?z x becomes 



- (f (a 3 + 2/3 2 a) - sin 2 * ) 4a - 2/3)) • 



and that of 



is - \ sin 2 * 2a - /3 sin 0 - 0, 



also 



= J(a 2 + /3 2 ) - ia 2 COS 20-20 + Ja/3 cos 20 - ^a/3 COS 20 



,/3 2 cos 20 + 20 



.-. 2y 2 = |(a 3 + 2a/3 2 ) COS 0 - 0 



and 



.rz 2 = i sin 2 * 2a - /3 cos 0 - 0 



