TABLES OF THE G (7, v)-INTEGRALS. 69 
« needed.! In these cases H (7, v) had to be derived from F (7, v) by 
(xiv.), and the logarithms of the factors were obtained from Vega’s 
10-figure tables. The methods applied should give both log H (r, v) and 
log F (r, v) correct to seven figures. 
On the whole we consider that the Table, if the calculations are not 
in error, ought to be correct to the number of figures tabulated. The 
calculations have been done with much labour and care, twice indepen- 
dently with 7-figure tables, and then again with 10-figure tables. The 
iatter investigation modified generally the seventh figure, and occasion- 
ally the sixth, but gave a much smoother system of differences. 
Logarithms of the functions and their differences were worked to ten 
figures and then cut off at the nearest figure in the seventh place, thus 
the recorded differences wre the nearest values of the true differences, and 
not the differences of the recorded logarithms. 
(4) With regard to interpolation formule for tables of double entry, 
we have been unable to discover much consideration of the subject, pos- 
sibly because hitherto such tables have been rather rare. We do not 
know of any formul, similar to those for interpolation on a curve, for 
interpolating on surfaces. The simplest formula, using second differ- 
ences, is : 
Uz, y— Uo, oO +x A Uo, ty) t yA'U,, 0 
+4 {x(a —1)A?u, + 2xy AAU, + y(y— 1)A‘74,, o A .  (xviil.) 
where A denotes a difference with regard to «, and A’ with regard to y. 
But if we consider w,,, to be the ordinate of a surface, and the figure 
to represent the xy plane of such a surface, then it is clear that, if P be 
the point x, y, and A, B, C, D, &e. the adjacent points at which the 
ordinates are known from the table of double entry, only the points 
A, B, C, D, J, and N are used by the formula ; and of these points, not 
equal weight is given to the fundamental points A, B, C, D, for C only 
appears in a second difference. If another point of the fundamental 
square other than A be taken as origin, we get a divergent, occasionally 
a widely divergent result. If we use only four points—A, B, C, D—to 
determine the value of the function at P, then we might take the ordinate 
at P of the plane which (by the method of least squares) most nearly 
passes through the four points of the surface vertically above A, B, C, D. 
We have then 
Ur, y=t (Up, ot My, op +Uo, 1+ %4, 1) +3(uy, om Uo, oF U1, 1— Uo, 1) (a—°5) F 
+ 4( Up, 1— Uo, ot U1,1—U1, 0) (y—'5) . (xix.) 
but by trial it has been found that this formula gives occasionally worse 
results than that for first differences, using only three points. To find 
by the methods of simple interpolation (with first or first and second 
differences) the points @ and 6,? and then interpolate P between them, 
generally gives a fairly good result ; but this result usually differs some- 
‘ For other investigations we have found, 
et? = et toe (4-20) = e201 + (w@—ao) + R(U@—Xo)? + B(—M)>+ —)s 
where ir, is the nearest value in Glaisher and Newman’s tables, to give e*” with 
great accuracy when four or five terms of the exponential expansion are used. But 
this method was more laborious than direct calculation when some 600 values were 
needed. 
2 See diagram on p. 70, 
