SUPPLEMENTARY CURVES. 



177 



In this case, if P M and A N be drawn perpendicular to the 

 given line u, the triangles A Q, N and P Q,, M will evidently 

 be similar, and we shall have 



PQ, : AQ, = PM : AN; 

 in virtue of which the preceding equation becomes 

 PM : AN = — m' (4). 



{b.) It is evident that the demonstration given above will 

 hold good when the proposed equation w = o is resolvable 

 into factors. Let it be resolvable into n factors of the first 

 degree, in which case it will represent a system of w straight 

 lines. Let PM„ PM„ PM,......PM„ and AN^, AN,, 



AN3, AN„ be the perpendiculars drawn from P and A 



respectively to these lines; then, by similar triangles, 



pa. : AQ, = PM, : AN,, PQ, : AQ, = PM, : AN,, &c., 



and therefore equation (3) may be written in the form 



P M, .PM^.PM, PM» , 



AN, . AN, . AN3 AN« ~ - "^ ^ ^' 



(c.) The last equation fails when each of the n straight 

 lines represented by the equation u = o passes through the 

 point A; since, in that case, each of the perpendiculars 

 AN,, AN,, &c. is zero. In this case equation {2) takes 

 the form 



M = A,3/« + A,2/«-ia; + A,3/''-V + A«aj" = o (6), 



and if we assume y = mx this becomes 



A,w'' + A,w"-^+ A^w''-^ + A„= o (7). 



Now if w,, m^, m^, nii, denote the roots of this equation, 



these will evidently be the direction indices of the system of 

 straight lines represented by equation (6), so that the equa- 

 tions of these lines will be 



y = m^x, y = m.iX, y = vinX (8). 



Let x' and y' be the co-ordinates of P, then, by the theory 

 of the straight line, we shall have 



2 a 



