302 



ENVELOPS. 



d^) l d<f> l db d<f> 1 dc _ 

 da db da dc da 



.(4). 



da db da 



da 



If the values of -y- and -y- from (4) be substituted in (3), 

 da da 



and then a, b, c, be eliminated between (1), (2), and (3), the 

 resulting equation between x and y will determine the re- 

 quired locus. 



This process may be rendered more symmetrical by suppos- 

 ing a, b, c all functions of a third variable, say t ; then using 



Da, Db, Dcfor-j-, -r , -^ respectively, we have instead of 



dt u/t cLt 

 (3) and (4) the equations 



dF dF n , , dF n A 

 Da + -==- Db + -,- Dc = 

 da db dc 



^Da + ^Db 

 da db 



da 



do 



, 

 dc 



dc 



= Q 



.(5). 



And the solution of the problem will be facilitated by the use 

 of indeterminate multipliers. Thus multiply the second of 

 equations (5) by \, the third by /*, and add to the first; 

 this gives 



\da 



dc 



dc 



(6). 



Now since \ and (j, are at present undetermined, we may 

 take them such that 



da 

 dF 



da 



~dB 



da 

 ~db 



(7), 



