,12HI 



If 7 uiul If' correspond to differeiil \iiliies of »« we see tlial for 

 tb'» orthog;onalitj it is necessarv and sufficient that 



I p{'>i)W{vmn)—M^i)'rW\\l = (20) 



If <f and If' correspond (o (he sauie n Ihe condition will be neces- 

 sary. So 7 i('t?), 7 ,(•*■)'••• ■ '^cing an orthogonal syslenj, (19) must l)o 

 valid for every pair of fniictions of Ihe system. 



Putting generally 



<fi{(i) = 'f'i . 7 '(«) = //( ' 7'i(/^) = "<' 1 7'i(^) ~^'i^ 

 we get from (19) 



p{a) (iVi yj — .Cji/i ) = p{b) {n,- vj - ujOi ) 

 for every pair of indices / and /". For the three indices /, /• and .s- 

 the equations become 



p{a) {xiiir—Xryi) = p{h){ui V,—U,.Vi ) 

 p{a) {x,ys — .*'.s.V,) = p{b) (UrVs- '/.st',), 

 p{a) (.Vsyi—xiys) = p{b) («.,t'/ — u,7,',). 

 Multiplying ihe first of Ihese equations by u,„ Ihe second l)y ii,. 

 the third by n,- and adding we get 



I "«■ •^•/ .v.' I 

 p{a) i Uf tVr y,- 1=0 



'*.s •«'•.s ys 

 In the same way, multiplyiiig by y,,, r, , r, we get 



V," •''■»• yi 



p{a) I Vr Xr y,. =0 



Vs Xs ys 



So, if /)(«) 7^ 0, it follows from this, by expanding Ihe determinanl 

 with respect to the elements of the first row\ 



Ui{Xrys -Xsy,) + Xi{y,.Us — ysnr) f yi{UrXs 'h''',) = ^i 

 r,(.tV//., -.';.,.(/,•) 4 ^iiyrl's-ysVr) f yi(v,X^—V,sXr)= 0. 



We now choose the indices /• and .v so that .r,.//^ — .i^sy,. ^ 0. Then 

 putting 



•Pr.V* — «s.Vr ' x,.ys—^syr ■ ' •'•»//« — x,y,. x,ys — .'•*.?/,• 



we gel for every value of i 



u, = axi 4- byi, 

 Vi = yjr, f ff//,. 



It is easy to verify that 



(«ff— ,?y) (.rrt/s - x.y,.) = n,.i\s — »<««', 



