( 118 j 
these three terms which have not the proper coefficients will be of 
little importance. 
These new variables are now to be introduced into equation (27). 
Yet it is not necessary to execute the substitution completely, for, as 
we have only to determine the sum of the coefficients of the squares 
ye", we may leave the coefficients of the products x, %, out of account. 
In the first place we have to express the old variables f in the new 
variables z We have: 
%, = 4), — BS, 
i as, Bf. 
4; = aje— BY, 
From this follows: 
vre) BDE a [uk 0 0 0... 0 
Ou see 8 eee 0 Mae a een ee 
i ) VO) AB 20 2.5). OTS x: Og. "OZ 
RS 40e en ue es | 0 0 0 ty a 
or fy (BY) = Of 2B Ff, BE yy BI 
In the same way we find for /;, 
Po (AR) = yo OA + opr 2B + ype ap... 
he fn OU nk Kn at aa 
In determining the products /./f, we shall always suppose that 
r’>vr, and so we shall integrate 7 between the limits O and +’; 
- between the limits O and x. In this way we get only one half 
of the quantity we have to determine. In order to find the amount 
contributed by the product ff cos(v'—r)tp to the coefficient of x. 
we have to distinguish three different cases: 
dist. O<r<v. Then the coefficient has the following form: 
| avdv? B2n+2r—v—v" cos (v' — v) Tp 
2nd. p< ld p'. Then: 
ante-po'—2r—2 nr -v- v eos (v'—v) Tp 
au rens Then: 
a2ntoyo'—2r—2 rvd’ cos (v'—v) T p. 
We have to seek the sum of these quantities, when 7 gets suc- 
cessively the value of all integers 1st. between 0 and v, 2°¢. between 
vr and #' and 34. between 7 and v. v has all values between 0 
and #', and v those between O and 7. We may write these sums in 
the form of integrals, if we put: 
EDU re == u Er =d Et 
al tand spie B: 
mans den ee 
