198 REPORT— 1882. 



The curve is exhibited in fig. 3, Plate I. 



(34) On oomparing the two methods a glance at the first and second 

 approximation curves in fig. 2, Plate II., and fig. 1, Plate III., establishes 

 a primd-facie case in favour of that adopted by Professors Thorpe and 

 Rlicker. In von Oettingen's method there is a considerable difference 

 between the two curves at the part affected by the additional measures, 

 and also at the points where the initial and upper point curves overlap. 

 In the curve obtained by the second method, the agreement is perfect at 

 the top of the scale, and very much better in the neighbourhood of the 

 overlap. The greatest difference occurs at the points which the previous 

 discussion indicates as the weakest — namely, the lower initial points. 

 The much closer agreement between the values of h',. — m', and of 

 Ji'^ — i)i' — 'w,., exhibited in fig. 3, Plate III., than between those in 

 fig. 3, Plate II., is also a point in favour of attributing the weight 10 to 

 the initial point curve. 



It is, however, possible to submit the differences in the neighbour- 

 hood of the overlap to a closer investigation. The values of h'' in Tables 

 , XXXI. and XLV. enable the correction of the initial points to be carried 

 a stage further. It follows from the investigation of the correction of 

 the initial points already given (p. 187) that, just as in the first appi-oxi- 

 mation, the process of combining the initial and upper point curves 

 changes ^ (i^) into (l> (i,.) + -nr,. ; so, in the second approximation, it will 

 change (p' (i,.) into (j)' (/,.) + •ct',.. Hence, if ^p' (/,.) is the difference be- 

 tween the second and first mean curves at the o-^^ initial point, 



^' (Q = f (V) + -'- 

 , By a method similar to that by which the correction Avas obtained in 

 the second approximation, the third correction is now found to be 



</." (i) = H,. - i + h'^ + -,V {f (n,„.) + i>' (u,„) + &c.} 



I Now, since in Tables XXXI. and XLV. the whole correction is taken 

 out — i.e. (j>2 («) — and not merely the difference between the first and 

 second approximations — i.e. (j)' (?;). 



i J^'r + tV [f (^h,r) + I'' (Mo„.) + &C.} = h" ,. ; 



and since <j> (v) = H^ — i„ 



and ^\. = ^P' (i) - f (i,), 



it follows that <}>" (i,) = h'\. - ■nr, - i/.' (?,). 



As -ra-,. + \p' (v) is the difference between the second mean curve and 

 the first initial point curve, this expression is exactly comparable with 

 that obtained in the first approximation. 



In Table LI. the values of (p" (r,.) are calculated according to both 

 methods. 



The values of <(>" (i,.) reqiiire the addition or subtraction of a constant 

 in order to bring the curves given by them into agreement at one point 

 with that obtained from the second mean curves. The cori'ected curves 

 could then be made to pass through the standard points. 



As, however, the values of ({>" (/,.) are ncai'ly the same at 100° and 

 116°, and as the resulting corrections ai'e very small, no appreciable 

 error will be committed if they ai^e considered as applied directly to the 

 standard curves. A third approximation to the corrections of the initial 



