408 HITCHCOCK. 



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

 dFp = 4>(p, dp), 



S'-a^-VpFp = SaV-[FaFp+ Vp^a, p)] 

 = 2Fa$(a, p) 4- 2VpFa, 



and writing j8 for p, we have twice the vector 



Va^{a, /3) + Vl3Fa (114) 



which is the polar vector of VpFp with regard to a at /3. 



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

 polar vector of VpFp takes at jS, and is perpendicular to ^. Also, 

 <E>(a, /8) is parallel to j3. If we agree to write 



n = VI3t, S'-aV-J^pFp = VI3t, (115), 



j8 being put for p after the differentiation, the parallelism of these two 

 vectors is expressed by 



VV^ttV^t = 



which, by a simple expansion, reduces to 



S/37rr=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 Vp'Fp + Vp^{p', p). If /3 is a double 

 axis, and /3 be written for p, the polar vector takes the form Syp'Vp-ir; 

 while if a be written for p and /3 for p', the polar vector takes the form 

 F|8t. The necessary and sufficient condition for /3 to be a triple axis 



is S^TTT = 0. 



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

 j8i be a double axis, we have, as already shown, ^22^33 — A23AZ2 = 0. 

 If i3i be written for p, the polar vector, by (62), becomes 



Fi3i)82(^22X3 + ^23.T2) + VjSMAs^Xz + ^33X2) (117) 



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



A22XZ + A23X2 = A 30X3 + ^33.T2 =0 (118) 



