t.'-( \na\Or8K* 



Substituting these values (8) in equ.itiMi, < j ,. u 

 ^(.e'oostf-y'Miiitf) 2 iKwl-i' >0 + y'coii0) 



+ / Mi0) i + :Jtf(*'oo0-y'Miii0) 



,*0)+<;-' 



which, being expanded and re-arranged* become* : 

 %Ac<*04 // 1000*0 



i ^oos^- 

 4- y\^ B" f ^ - 2 /T in ^ cxw 6 + coil 1 0) 



J0C080 U0) 



2a8in0-H2^ f ooe^)--C r -0. . (5) 



transformed equation (:">) will l>e free from the term 

 taining the product x*y' if 6 be so chosen that 





illy, if 



Moreover, it is always possible to choose a positive acute 

 angle B so as to satisfy this last equation whatever may be 

 the numbers represented by -4, /?, and //. 



II M chosen so as to satisfy ecpi.it i<*n (ti), and having 

 substituted the values of sin0 and cos in equation 



thilt IMJllati.Mi iv.lllrrs tn 



i -'+-B i y a ---j/;v-fjf t y+c-o f . 



herein A\ ff< represent the new coefficients) 



and thm fore represents a conic section with its axes parallel 

 ue new coordinate axes. But equation (7) represents 



