Rev. B. Bronwin on certain Definite Multiple Integrals. 375 



/»/* dxdy ... _ otS ... /»/» dxdy ... . . 



JJ '"~W Z ^~ ab..JJ '" R?~ 2 ' ' * ( j 



In the next two examples, by way of distinction, let #, y, &c. 



be changed into ax, a,r, 6y, 6y, &c. ; and let there be three 



variables, 



« § y dx dy dz 



fffi 



{(g-ux)2 + (k-fy)*+{k-yz)*}t 

 _*§y rrp abcdx dy dz 



~ aTcJJJ { {g - axf + {h- byf + {k - czf}* 



= rp >Sydxdys/l-x*-f +Ac , Bc2 + &c 



aSydxdy V\ — x 2 — y 2 

 {(g-axf + (k-byf-tk z }i 



SJ\ 



(2.) 



the equation of limits for the first member being x 2 +y 2 + 2 2 = 1, 

 that for the second x* + y' 2 =l. This result is obtained by de- 

 veloping into series relative to z, then integrating for this 

 quantity, and lastly diminishing c without limit, the quanti- 

 ties A, B, &c. being finite. In the second member it must 

 be observed, that since c=0, a= VV— 7 2 , b= VSP—y 2 . 

 If we differentiate (2.) for (&), we have 



M 



(fc — yz) dx dy dz 



{{g-«x?+{h-Sy?+(k-ys?Y 



~m 



kdxdy V ' l—x 2 —y 2 



{{g-ax)* + {h-by)* + Ji*}% } 



(30 



By integrating the first members of (2.) and (3.) relative to z, 

 we should obtain very singular results. 

 Let 



R={Cg-*)H (*-#+(*-*)«}*> v=J£f*^v±, 



U the same integral when «, £, y are changed into a, b 9 c. 



Make 



v Vto — tL ^ J' =sm u, — & — sin v. 



R S{g-xy + {h-yf 



Then 



x=g— Rsinwsinv, y—h — Rsinwcosw, z=k— Rcosm, 

 dx dy dz= — d R sin u du dv ; 

 or rather 



