fil 



we should find, by the foregoing Section of this Abstract, and 

 especially by the equations (37) and (38), a result of the form, 

 (s^ - n^s^ + n'^s - n"'^)i = o-'k, (52) 



where a is a new characteristic of operation, such that 



(7'=(T^-5<r + «UwV-s) + n", (53) 



and that, therefore, 



OK = s\ + sS . Wa r. OK - S . mmV. aa'S . aaK ; (54) 

 so that the solution (41) of the linear equation (39) is included 

 in this more general result, which gives, for any arbitrary va- 

 lue of the number s, the symbolic expression : 



((T + 5)-i = (s3 - n^i^ + n'-s - n'Ya. (55) 



Hence the condition for the non-evanescence of the expres- 

 sion (50), or the distinctive character of the permanent axes 

 of rotation, is expressed by the cubic equation, 



s''-wV + n'2 5-w"^ = 0. (56) 



The inequality (43) shows immediately that this equation (56) 

 is satisfied by at least one real value of s, between the limits 

 and n^ ; and an attentive examination of the composition 

 (35) of the coefficients of the same cubic equation in s, would 

 prove that this cubic has in general three real and unequal 

 roots, between the same two limits; which roots we may 

 denote by s„ 5,, s^. Assuming next any arbitrary vector k, 

 and deriving from it two other vectors, k' and k", by the for- 

 mulae , /r. ' " (f^'7\ 

 2 . ma F. aK = k' ; - S . mm V .aaS . aaK=K ', {pi) 



making also 



ti = Si\ + SlK + K , -^ 



li = S2^K + S.2K-\-K', > (58) 



tj = S3^K + S^K' +k";J 



we shall thus have, in general, a system of three rectangular 

 vectors, .„ h, '3. i" the directions of the three principal axes. 

 For first they will be, by (54), the three results of the form 

 ct'k, obtained by changing s, successively and separately, to 



