i8o Journal of Agricultural Research voi xii. No. 4 



and (x-c)-+{y-dy- = R' 



Z=2k. 



Draw the line from a point {x^, y^, O) on the first circle to a point {x^, y^, 2k) 

 on the third. Its equations are 



from which 



x-x^ _ y-yi _ z ^ 



^2~% y2~yi 2fe 



2k(x—X,) + z(x,—c) 



- = x^ — c. 



Z 



2k(y-y^) + z(yi-d) 



■■y^-d. 



Since 



{x,-cy+{y,-df = R\ 

 we have, after simplifying, 



^\{x-x,f+{y-y,y]+Akz[{x-x,){x,-c) 



+ {y-yd{y-d)]+z\{x-cf+{y-dy-R']=o. (loi) 



This is the equation of a cone with vertex at {x^, y^, O) and passing through 

 the third circle. 



In the same way, find the equations of the line from (x^, y^ O,) to 

 («3. yst k) on the middle circle 



x-x ^^ y-yi _z ^ 

 ^3-^1 yz-yi k' 



k(x—x^) + z(x^ — a) 



z 



Hy-yi)+z(yi-b). 



■ ^'3 — <i> 

 -ys-^- 



Since 



(x,-ari-(y,-b)2 = (^fj' 

 we have, after simplifying, 



k\ix-x,y+(y-y,r]+2kz[(x-x,)(x,-a) 



+ {y,-b)(y-y,)]+2'^(x,-ar+(y,-by-~j=0. (102) 



When equations (loi) and (102) are multiplied out, it will be seen that 

 x\, fi always enter in the form x\ + fj^ = R\ By substituting R- for 

 0^1+ y^i in each, the equations are of the forrr 



Ax,+ By,==C, 

 A%+B'y, = C'. 



