SOME PROPERTIES OF PROJECTED SOLIDS. 227 



then, by substitution, we may show, I denoting the 

 integral 



ccc 



<j> (x, y) dx dy dz, 



(x— a)cos«+(y— 5)co8/3 

 cosy 



that 



I z =cos*7l. 



As another example of the use of the substituted 

 coordinates, suppose that between any two points of the 

 primitive solid any arbitrary curve be drawn lying on the 

 surface. Let s be the length of this curve, and ds an 

 element extending from the point x 3 y, z to the point 

 x + dxy y + dy y z -f dz. Let s z , s y) s x be the lengths of the 

 curves on the three derived solids passing through the 

 points corresponding to those through which the curve 

 on the primitive solid passes; then, ds z> ds yi ds x being 

 elements of these curves, we shall have 



dst^df+dy' + dg 1 ; 



by substitution this becomes 



ds\ — dx % + dy z + cos 2 7 (dx cos u + dy cos fi + dz cos y) % 



and in like manner may be obtained 



ds % y = dx z + dz 1 + cos z f2(dx cos * + dy cos /3 -f- dz cos y) 2 , 

 dsl= dy z + dz 1 + cos*a (dx cos ct + dy cos (2 + dz cos 7) 1 . 



By addition we obtain 



ds\ + ds z y + ds x = 2 (dx z + dy x + dz z ) 



-f (dx cos a + dy cos /3 + dz cos 7) \ (5) 



Let <j) be the angle between the primitive axis and the 



