(iUADRATIC VECTORS. 409 



By (9) these equations are equivalent to 



^22S/3l^2P + ^23S|83/3lP = As2SPi^2P + AzzS^z^iP = (119) 



Let 7 and /3i be supposed unit vectors; and let co and Ci be two con- 

 stants defined by 



^22F,3i/3o + A23V^3^i = c.y, A32V^i02 + AssVpslSi = C37 (120) 



The equations which determine the tangent plane then become 



C2S7P = C3S7P = (121) 



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



F^l(c2/32 + C3|33)S7P (122) 



The required vector tt is thus given by 



T = C2)82 + C3/33 (123) 



The direction of a is Vj3iy, which is certainly determined by 



c'2V^liAo.l%l32 + A03VI33I3,) + c'sV^l(AsoJ%02 + ^33F^3^l), (124) 



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

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

 To find r we have, by the rule, to write /3i for p, (instead of for p'), 

 in the polar vector, which then becomes 



VfiiFp + Fp^i(.4i2.r3 + Anxo) + Fp(c2^2 + C3j83)S7P (125) 



On writing a for p, 1S7P vanishes; and the remaining terms are at right 

 angles to /3i. It is therefore obvious that, in (124), we may neglect 

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



(^r - ^lS^l)[(c'2^22 + c'zAz2)^2 " (c'2^23 + c'2^33)^3] (126) 



By dropping the component along /3i, we see that we may use, instead 

 of the true value of a, the simpler vector 



(c'2^22 + c'3^32)i32 - (c'2.423 + c'zAsz)^z (127) 



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



