408 HITCHCOCK. 



when applied to VpFp at a point on the element /3. Now if 

 dFp = ^{p, dp), 



S-aV-VpFp = SaV-[VaFp+ Fp$(a, p)] 

 = 2Fa$(a, p) + 2VpFa, 



and WTiting /3 for p, we have twice the vector 



Va^ia, 13) -\- JWa (114) 



which is the polar vector of T^'pi^p Math regard to a at j8. 



Returning now to the direction rj, this is the direction which the 

 polar vector of VpFp takes at /3, and is perpendicular to (3. Also, 

 <l>(a, /3) is parallel to /3. If we agree to write 



V = VlBir, S^aV-VpFp= V^T, (115), 



/3 being put for p after the differentiation, the parallelism of these two 

 vectors is expressed by 



VVIStV^t = 



which, by a simple expansion, reduces to 



S^TTT = Q, (116) 



which is both necessary and sufficient that g shall be independent of X. 

 We may sum up the foregoing investigation of triple axes in the rule : — 

 Let the polar vector of VpFp be T p'Fp + J'p^{p, p). If /S is a double 

 axis, and jS be written for p, the polar vector takes the form Syp'V^ir; 

 while if a be written for p and j3 for p', the polar vector takes the form 

 V^T. The necessary and sufficient condition for /3 to be a triple axis 

 is S/Sttt = 0. 



19. It now becomes a simple matter to apply this rule to (61). If 

 /3i be a double axis, we have, as alreadj^ shown, ^422-133 — ^23^32 = 0. 

 If jSi be WTitten for p, the polar vector, by (62), becomes 



Fi8i^2(^22a;3 + ^23.1-2) + V^i^3{A32X3 + AszX.) (117) 



The normal direction is thus the normal to the plane determined by 



^22^:3 + ^23.^2 = ^432.T3 + ^33.T2 = (118) 



