234 
DR. W. F. SHEPPARD ON 
Then, on the assumption of normal correlation of errors, the frequency of joint 
occurrence of these 0’s is proportional to 
exp —|-P, 
where P is a homogeneous quadratic function of the 0’s. We want to prove that 
(i.) the (j> s are the partial differential coefficients of \P with regard to the 0’s, 
and conversely ; 
(ii.) P = 0000 + $101 + $202 + • • • + Q/ ( f>l > 
(iii.) P = '0- o>o $o 2 + 2 i Ao, 1 0o0 1 + 1 A'i,i$i 2 + ••• +Vo,z$z 2 > where \fs fg is the m.p.e. of y f and 
y g ; and similarly 
(iv.) P = tt 0 ,o0o 2 + 27Ty, 1 0 o 0 1 + x 1 , 1 0 1 2 + ... +7 T u ,<pi\ where tt /3 is the m.p.e of u f and u g . 
2. Suppose that 
P = Oo, 0$lf + 2«0,10 O $1 + «l, l$l“ ?0p ; 
and let us, without making any assumption of conjugacy, write (f — 0, 1, 2, ...l) 
yf = Uf^ 0 U 0 + Uf pq + 2 'M 2 +... + Uf t 
<pf = error of y f 
— Uf' Q0o + Uf' j0j + Uf' 2 0 2 + • • • + Uf' 
= HP/dOf. 
Then, writing the subscripts in the order f, 0, 1, 2, ... I, 
P = a f,fQf + % a f, o$/$o + a 0, o<V +•••+<*;, /0/“ 
= < P/luf'f+Q , 
where $ does not contain 0y. 
3. The mean value of 0^0 ? is NjD, where 
N g = [ j [ • • • 10/0 ff exp— \P . d0 o c?0j ... dd h 
D = j jj... |exp — \P . dQfdOodOi ... d6 h 
the integration being in each case from — co to oo. If we write 
= 0// \/ a f,f> 
then 
N g = |Jj ... j*0-0 ? exp—-g-i/r 2 . exp—. d\p-d0 i) d0 1 ... d0,. 
(a) First, suppose that g is not = f. Then, integrating with regard to \}s, 
Ng = 0. 
