HiNTON — Cayle]/s Formulm of Orthogonal Transformation. 61 



and represents a plane pair whose constituents are to a certain extent 

 indeterminate, the plane pair is derived from all those perpendicular 

 planes, the corresponding sums of whose direction cosines are the 

 same. By "perpendicular," I mean perpendicular like/e and M, not 

 normal, like the planes oiji and hj. 



Any of these plane pairs will turn a vector round in the plane of 

 AjAo, and in the perpendicular plane. 



jSTow, consider the w form of the plane X1A2, 



w {yji + ahj + ^ik + y'hh + a'ih + fi'jh) ; 

 we obtain the plane pair 



(y - y') ^j^ + (a - a') whj + (/3 - /3') wih. 



This plane pair rotates a vector in the plane of AAzj and in the per- 

 pendicular plane by a right angle in the direction h to h. 

 Hence the product 



[cos e + sin ^ { (y + y')ji + (a + a') hj + (/3 + /3') ik}'] 

 X [cos ^ + sin 6 { (y - y') wji + (a - a') ijikj + {/3 - (3') mk]^ 



will rotate a vector by 20 in the plane of XjAo; and its projection on 

 the perpendicular plane will be unaltered. 



If, instead of taking both angles equal, we take different angles 

 6 and <f>, we get the rotation of amplitude ^ + <^ in the plane of 

 y, a, y8, y', a', /3', and ^ - <^ in the perpendicular plane. It is most 

 convenient to take n, I, m, n', I', m' as the direction cosines of a plane, 

 instead of y, a, /3, &c., and denote the sums n + n', &c., by y, a, /?, 

 the differences n - n', &c., by y', a', ji' . The most general rotation is 

 given by letting both angles be different, and taking two plane pairs 



yji + akj + ^ik^ and y'co;"? + a'<jdkj + (3'u)ik, 



which I will call tti and unr'i, where y and y', &c., are unrelated, and y 

 and y' stand respectively for n + n' and N - N\ and so on, where Imn, 

 I'm'n', LIIN, LM'N' are the direction cosines of two independent 

 planes. 



In order to present the multiplication in a conspicuous form, I 

 write out the multiplication table of the quaternions, with the coeffi- 

 cient adjacent. By assigning values to ^, 0, & ., 4>', y, y', c, c', &c., 

 we can find the combined rotation equivalent to any pair of rotations. 

 For cos 6 I write ^1, and sin 0, 0. It will be found that wji and 7'/ are 

 commutative as well as any other A and B pairs. The multiplier 

 occupies the two columns to the left. 



