Series to Frequency Distributions in Space. 381 



Then F(«, a!, J3, 7, .«/) 



- Xo+Xl l^7' /+X2 ^72 tTtTI) 3 ' + • • • • 



- Y ° + il lTV +ls i.2. 7 ( 7 TTT' + ' • • ' 



orF=2X s ./ = 2W-'. 



Now F(a, /3, 7, ,r) satisfies the equation 



ff(l-*)g + ( 7 _( a +£ + l)*)g_«/32=0, 



Multiply this by ?/'' and sum with respect to s' ; writing 

 z for F(«, a', /3, 7, .r, //) we obtain 



X (l-.v)^+(y-(u + (3+l).v)^-al3 : + (l-.v)ls'^ : (X,f') 



-Oits'X,/' =0. 



9y 



But^5i.=2s'X s ,//, 



(2) 



-d— )g+(7-(«+^+D-)|;-^+(i-)^|)-^|=o ; 



or finally z satisfies the two partial differential equations 



*(i-)g+{7-(-+^+i>}||+ya-) d ||-«y|-^=o • (i) 



§ 5. 77*£ moments of F x (a, « b /3, iy). We imagine an 

 ordinate of magnitude z(s, s') erected at the point 

 ,r = cs, y = cs', where c, <"' are constants to be determined in 

 the fitting. LetV be the volume of the polyhedron of which 

 the tops of the ordinates are vertices, /, e. a polyhedron 

 made up of elementary prisms on base cc', and of height 

 z(s, s')j and letpw denote the (t, t')t\\ product moment about 

 the centroid vertical, the elements of volume of the polyhedron 

 being concentrated alonu the ordinates. Loty>'«' be the corre- 

 sponding moment about the planes 



ff=— c, !/=-c'. 



