DR MATTHEW STEWART'S GENERAL THEOREMS. 



centre of the porismatic circle, p its radius, and K the inclination of the arbitrary diameter 

 to the axis of a:. Then we shall have 



A ' } (1) 



Again, the equations of the porismatic circle and its arbitrary diameter will respectively be 



y fi=(x a)tanK (3) 



Denote, for the moment, the points X, Y by tfy' and x"y" : then from (2, 3) we get their 

 values 



z 1 = + ^)cosK a^'=a p cos K 



Whence 



And therefore 



XZ + YZ > = 2{* s +y-2a*-2/8y + a + |S I + p} .... (4) 

 which, as in the former solution, is independent of the value of K. 



With the elements (1) and (4) form the equation of the porism ; arrange the terms in 

 reference to the arbitrary quantities x and y ; cancel S a m . (x 2 + y 2 ) and equate to zero the 

 remaining co-efficients in respect of x and y ; then there are given the three following condi- 

 tional equations for a, /3, and p. 



o 2 + . . . . + a m a m =Sa m .a ......... (5) 



( S i! + ---- + a m /3 m =Sa m .(3 ........ (6) 



....a m (a m *=p m *)=8a m .(a'+p + f) .'. . (7) 



From (5), (6), we have 



Sfr.aJ ^BSg&J 



Sa m Sa m 



which are the co-ordinates of the centro'id ; and which point is, therefore, the centre of the 

 porismatic circle. Also, from (7) we have 



That this value is always real, may be readily shewn by actual substitution of the values 

 of a, /B ; but it readily follows from transforming the origin to the centro'id, whose co-ordi- 

 nates are a /3. For since 



o m .{(a-a) 2 + (^-/3) 2 }=0, 

 the value of p 2 becomes 



So. 



and every term of this being positive, their sum must be so, and the value of (> real. 



