116 
MR. W. F. SHEPPARD ON THE APPLICATION OF THE THEORY OF 
of /L + mM are distributed normally with this mean square'" about a mean value 
/Lj -J“ wilVlj. 
Next take the more general case. Since the distributions of Q and of E, are 
independent and normal, the distribution of + rE is normal. Again, since 
the distribution of P is independent of the distributions of Q and E, it is 
independent of the distribution of qQ + rE ; and therefore, since the distribution 
of P is normal, the distribution of pP + qQ + ^'E is normal. Proceeding in this 
way, we see that if the distributions of L, M, N, . . . P, Q, E are mutually inde¬ 
pendent, and if each distribution, taken separately, is normal, the distribution of 
/L “h + nN + . . . + pP + + ^’P^ is also normal. 
We might have obtained this result from the statistical equation of the normal 
curve (§ 5)'; Let L-Li = T/, = M', N-N, N',.... Also let S ...) 
denote the mean value of • • • , and let denote the mean value 
of (7L' -f mM' -h wN' + . . Then, since the distributions are independent, 
S . . .) = S (L'‘^).S (M'^).S (N'^).... Also, by § 5, S (L'-*" *) = 0, and S 
l-s 
= a~ 
2'(s 
and similarly for M', N', . . . . 
Hence we see that— 
(i.) Every term in the expansion of (/L' -P mM' -p nN' + . . must contain an 
odd power of one at least of the quantities L', M', N', . , . ; and therefore, by taking 
the mean, X 2 s_i = 0 ; 
(ii.) X, = l~u~ -p m^6‘ + n-c~ + . . . 
(iii.) Xo, = mean value of {lU + mM' -p nN' -p . . .Y" 
= -^2 ... |2.|2g%... Sf(^L7"j.S {(mM'rj.S [{«N'y'»l .., 
(the summation being made for ail positive integral values of a, /3, 
y, . . . satisfying the condition a-p^-Pyd--- - — s) 
— vw 
is 
2»!s • • • 
la |2/3 |27 . . . 
i'S’ 
py. . . 
. . . Pt- a-“. 4= . . . 
2 “|« •P\I3 2^7 
{ycrY.{iifh-Y.{n-y-)y ... 
^2,S 
s 
and therefore, for all positive integral values of /r, 
N +2 — {y + 1) kA/,. 
* The expression “mean square” may generally he used, without eonfitsion, to denote the meati 
square of deviation from the mean, 
