35,2 



REPORT 1869. 



These arc taken as fundamental formuliB by Schellbach, and the following 

 four groups deduced from them : — 



dxd,x =6(0,2^)9 (2x, 2,), ^ 

 e^xd.jv=dio,2,')8,{2x,2v), I 



e(o)d,{o)=e(o,2ry=dQ7r,h'y, 

 en{o)d,(o)=id,(o, ivy. 



ho'' hx^ — go' r/.y- = 1 , 

 7io''fu^ + r/.i 

 go'f£-+ hx--- 



fx-=i, -. 



o ■> I 



r/x-=go-, V 

 hx"=^liO'. J 



{a) 



(y) 



From (5) and (8), using the first of equations (a), we have at once 

 and similarly 





Oo' 



fli(-^'+;/)Oi('^--y) 



Ox' Qy- 



=f'^-'-ff, 



> 



(0 



n 2 flX'-' + y) fi;(-V — 1/) TWO 1 



' ex-67f -^ 



O.fi . -^ — ■" '^ 11/ = 1 + nx- qir. 



OX mj 



Several of these methods and formula! appear to he due to Krusemarck. 

 Section 9. — The following formulaj are derived ])y multiplication from 

 (])... (12) of last section :— 



2dx d.j>:6yd.,y=dod..o{Q{x + 7j) Q.ix-y)-^Q {x—y)Q.lx-\-y)} 

 2B,xQ,x d,yQ,y=doe.fl{B{x+y) d,(x-7j)-d (x-7j) 0,(a-+y)} 

 20,xd,x ey e,i/=eod,o{e,(x+y) e,(x~y) + 9Xx-y)d.Xx+y)} 

 2dxd,xd,7jdjj=eod,o{d,{x + 7j)0.p--7j)-0,{x-7j)dX.v+y)} 

 2dx d,x(Kyejj=e.fiO,o{9X^ + 7j)d (x-y) + d(x+y)B,(-v-y)} 

 2d,xd,v dy 0,7/ = e.M,o{0^{x + 7j)d (x-y)-O (.r+.y)e/.r-y)} 

 2d,vd,x By B.jj=Bo Bfl{Q,{x+7j)B.Xx-7j) + BX-r-7j)B.ix^y)] 

 2Bx B,x B,ydjj = 6o a,o [9,(,r+^)9^(.r-y)-eX.r-y)9,(,r + 7/)} 

 2dx 9,x By 6,y = Bo B,o{dX^v + 7j)B (a'-y) + 6,(x-y)B (.r+y)} 

 20^xd.x B^7jBjj = Bo 6^o{B,{x-y)d {x + 7j)-B,(x+y)6 {x-7j)} 



(1) 

 (2) 

 (3) 



('^) 



(■'5) 

 (0) 

 (7) 

 («) 

 (9) 

 (10) 



Let 



2T=-.l-fx' f/ = 9o=!(£iJM^i:i^\ 



■ "^ Ox' By' 



