288 M. Poinsot on the Percussion of Bodies. 



Corollary II. 

 On the centre of maximum percussion. 

 20. To find the coordinates x and y of the point D where the 

 percussion Q is a maximum, we have merely to form the two 

 equations 



^=0, ^=0, 



dx ' dy 



which give 



u''a[oL''x'' + ^y"" + «'/3') - 2u^x{o?ax + ^by + u^jS") = 0, 

 /S'b (« V + ^y2 + «'2^) _ 2/3^2/ [oc^ax + ^by + u^^^) = 0, 

 whence we immediately deduce the proportion 

 X -.y^^a -.b. 



This shows, in the first place, that the required point D lies in 

 the direction of the line C G joining the centres of impulsion 

 and gravity. Substituting, therefore, in either of the preceding 



equations, for y its value —x, and subsequently for x its value 

 ^y, we shall have, for determining x or y separately, the qua- 

 dratic equations 



(a^flS + ^2^,2)^2 ^ 2oL^^ax - cc^fi^a^ = 0, 

 (a2«2 + ^2^,2)^2 ^ 2c,^/3''by - u^l3^^ = ; 



these equations are exactly similar ; the first resolves itself into 

 the second by simply changing x and a into y and b, and vice 

 versa. 



21. But, since the required point D is on the given line C G, 

 let us make the distance i; = DG between the required point 

 and the centre of gravity G our unknown term ; by making 



CG= \/a^ + b^ = ll, 



we shall have the proportions 



V : H=;r : a=y : b, 



whence 



a , b 



x= Y^v, a.ndy= jfV. 



Substituting these values in the preceding equations, we have, 

 to determine v, the equation 



^ +''^" «V -1-/3262 •« ^ ci^a^ + /3^/j'' ' 

 but, A being the distance between the spontaneous centre O and 



