262 EXAMPLES OF MAXIMA AND MINIMA. 



By subtraction 



o'P (kp + ] Pn? (kp + /} cV (kp + t\ 

 \ r aU \ b-mj \ r c*n) A 



* * 



Now Tcu 4- '-57 , Tcu. + o- > an d ku, + -~- are of equal value 

 or* 6m c AI 



by (4) ; and this value cannot be zero, because then by (12) 

 we should obtain the inadmissible results x = a, y = /?, 3 = 7. 

 Hence dividing out we have 



a 2 / 2 6 2 w 2 c 2 /i 8 



&a* + r* kb* + r* kc* + r* ~ ' 



This quadratic will give two values of r 2 , one will be 

 the maximum value of r 2 and the other the minimum value. 



The product of the values of r 2 will be 



and TT times the square root of this product is the area of 

 the curve of intersection of the ellipsoid and plane ; hence 

 taking the positive value of the square root we have for 

 the area 



iralc (o'P + Vm* + cV - p*) */(? + m* 



* + n*)~] 

 J* 



C. Find the maximum or minimum value of u when 

 u = x?y 3 z*, and 2x + 3y + z = a. 



Result. f-J is a maximum value. 

 7. Find the minimum value of u from the equation 



the variables being connected by the equation 

 ax + by + cz + = k. 



Result, u = - g j-j s . 



a* + b 2 + c 2 + 



