QUADRATIC VECTORS. 409 



By (9) these equations are equivalent to 



A22S^i^2P + Ao^S^s^rP = AsoS^i^oP + AszSfispip = (119) 



Let 7 and j3i be supposed unit ^'ectors; and let c^ and Cs be two con- 

 stants defined by 



^22F|8i/32 + A2sl%l3i = C27, As^VlSi^, + AisV^s^, = c^y (120) 

 The equations which determine the tangent plane then become 



C2S7P = CzSyp = (121) 



and the polar vector for /3i becomes, by (117), 



F/3i(co|3o + C3/33)^'7P (122) 



The required vector tt is thus given by 



TT = C2/32 + 03)83 (123) 



The direction of a is Vl3iy, which is certainly determined by 



c'2F/3i(.Ioori3i/32 + ^23F/33/3i) + c'zVfiMs2l%0.^ + AssV^s^O, (124) 



since, under the present hypothesis, the four ^'s are not all zero; 

 C2^ and Cs^ being any two new constants such that (124) does not vanish. 

 To find T we have, by the rule, to write jSi for p, (instead of for p'), 

 in the polar vector, which then becomes 



Vl3iFp + rp/3i(.4i2a-3 + ^13.1-2) + Fp(c2/32 + C3^3)S7P (125) 



On writing a for p, Syp vanishes ; and the remaining terms are at right 

 angles to jSi. It is therefore obvious that, in (124), we may neglect 

 any component along /3i. Multiplying out, (124) gives 



W - /3l^'j8i)[(c'2/l22 + C'3A32)^2 " (c'oAos + c'2^33)/33] (120) 



By dropping the component along /Si, we see that we may use, instead 

 of the true value of a, the simpler ^'ector 



(c'2^22 + c'3^32)/32 - (c'oyl23 + c'3^33)i83 (127) 



The required vector r is the result of writing the above vector for p in 



