of Least Pressure to the Theory of Resistances. 97 



li l 'ts i 4v l tt dy l +u 1 m tx l = 0' 



u 2 ' 8 x 2 + u 2 " 8 y 2 + u 2 " ! 8 z 2 = 



&c. = 0J 



Cos otj sin at, 8 a, + cos ft sin ft 8/3, + cos y! sin y 1 $y l = ' 

 Cos a 2 sin a 2 8 a 2 -f cos ft sin /3 8 ft + cos y 2 sin y 2 8 y 2 = 

 Cos a 3 sin a 3 8 « 3 + cos /3 3 sin /3 3 8 ft + cos y 3 sin y 3 8 y 3 = 

 &c. = 0J 



Hence, adding the above equations, having first multiplied 

 equations (1'.) by the indeterminate quantities A! A 2 A 3 re- 

 spectively; equations (2'.) byB 1 B 2 B 3 ; equations (3'.) by 

 Aj A^ A 3 A 4 , &c. &c. ; equations (4/.) by fx { \l 2 jx 3 ft, 4 , &c. &c. 



(3'.) 



(4'. 



! 



rfP 



^P=^ 



-r~{] + (A 1 + B ] y— B 2 z)cosa + (A 2 — B 1( r+B 3 z 

 a> x 



cos /3 + (A 3 +B 2 o:-B 3 3/) cosy} 



+ P(-B! cos/3 + B 2 cosy)+Aw' |.8r. 



+ {g{l+(A 1 + B li /-B l2 )cos« + (A 2 -B 1 ^ 

 + B 3 z) cos /3+ (A 1 + B 2 j;-B 3 ?/) cos y} 

 + P(B 1 cosa-BaCOsyHAi/'I.Sj/. 



1 az l 



-f B 3 z) cos/3 + (A 3 +B 2 ^-B 3 j/) cos y} 



+ P(-B 2 cos« + B 3 cosft + Aa'''}.8z 



-p{(A 1 + B 1 j/-B 2 s + j^cos«)sin«8a + (A 2 -B 1 ^ 

 + B 3 z + |*cos/3)sin/38|3+(A 3 + B 2 .r-Bg > y 

 -f p cos y) siny 8y I. 



Now, let the indeterminate quantities A t A 2 A S , B l B 2 B 3 , 

 X Xo Xo ... f^! ft 2 p 3 ... be taken so as to satisfy the equations of 

 condition (1.) (2.) (3.) (4.); *, y x z 19 x 2 t/ 2 z 2 , x 3 y 3 z 3 ... a, ft y,, 

 a 2 £2 72» a 3 ^3 73 •*• ma y tnen ^ e cons idered independent vari- 

 ables-, and by the condition 



3"P = a minimum, 



Third Series. Vol. 5. No. 26. ^. 1834. O 



