284 PROFESSOR TAIT ON THE ROTATION OF A 



which has been assumed all along. In fact, we see at once that 



wW + xX + yY + zZ = 



identically, which leads to the above integral. 



These equations appear to be worthy of attention, partly because of the homo- 

 geneity of the denominators W, X, Y, Z, but particularly as they afford (what does 

 not appear to have been sought) the means of solving this celebrated problem at 

 one step, that is, without the previous integration of Euler's equations (§ 23). 



A set of equations identical with these, but not in a homogeneous form (being 

 expressed, in fact, in terms of /c, X, /x, v of § 13, instead of w, x, y, z), is given by 

 Cayley {Camb. and Dub. Math. Journal, vol. i. 1846), and completely integrated 

 (in the sense of being reduced to quadratures) by assuming Euler's equations to 

 have been previously integrated. (Compare § 27.) 



Cayley' s method may be even more easily applied to the above equations 

 than to his own ; and I therefore leave this part of the development to the reader, 

 who will at once see (as in § 27) that %,, 33, € correspond to w JS &> 2 , &> 3 of the >; type § 23. 



29. It may be well to notice, in connection with the formulae for direction 

 cosines in § 13 above, that we may write 



& = -r\ a (w 2 + x 2 — if — z 2 ) + 2b (xy + wz) + 2c (xz — wy) , 



28 = p 2ft (xy — wz) + b (w 2 — x 2 + y 2 — z 2 ) + 2c (yz + wx)\ , 



= tt\ 2« (xz + toy) + 25 (yz — wx) + c (w 2 — o. 2 — y 2 + z 2 ) . 







These expressions may be considerably simplified by the usual assumption, 

 that one of the fixed unit-vectors (i suppose) is perpendicular to the invariable 

 plane, which amounts to assigning definitely the initial position of one line in the 

 body ; and which gives the relations 



6 = 0, c = 0. 



30. When forces act, y is variable, and the quantities a, b, c will in general 

 involve all the variables w, x, y, z, t, so that the equations of last section become 

 much more complicated. The type, however, remains the same if y involves t 

 only ; if it involve q we must differentiate the equation, put in the form 



and we thus easily obtain the differential equation of the second order 



if we recollect that, because q~ 1 q is a vector, we have 



S.rt = (<rt)>. 



