20 Archdeacon Pratt on the Determination of the 



A to the mean ellipse ; r A a a normal through A to the vari- 

 able ellipse ; s A a line through A making a small angle z 

 with r A a. 



Now, if the earth had its mean form, the plumbline at A would 

 hang in the line nAa, the normal to the mean ellipse. (N.B. 

 As A does not coincide with a this would not be true, unless 

 A a were a small quantity ; but as it is small, the error in sup- 

 posing the direction of gravity at A to coincide with the normal 

 from A to the ellipse is of the second order, and may therefore 

 be neglected.) But it actually hangs in the line m A. Hence 

 the angle m A n is the deflection produced by local attraction at 

 A, and equals t. The angle n Ar, between the normals through 

 A to the two ellipses, = n(v — V), where n is a known quantity, 

 and V the value of v for the mean ellipse (see ' Figure of the 

 Earth/ p. 129). Hence by applying to the observed latitude of 

 A the quantity t + n(v — V) — that is, the angle mAr, I make the 

 point a on the variable arc correspond with A on the actual arc, 

 and the points /3, 7, & are at the same distances from a along 

 the variable ellipse as B, C, D are from A along the measured 

 arc. This, then, was the value which I gave to x, the correction 

 of the latitude of A. I at first, however, added another correc- 

 tion still, viz. an arbitrary small angle r A s or z, with the same 

 view for which Bessel used his x, viz. to adjust the small errors 

 introduced by observation and measurement. In this case x 

 would have equalled t-\-z-\-n(v— V). But I had no sooner 

 substituted this in the sum of squares of errors and differentiated 

 with regard to z, than I perceived that the resulting formula 

 would give me, not z alone (which I wanted), but z in insepa- 

 rable company with t, thus t + z, and that therefore when I 

 eliminated z I should be eliminating t also, and my end would 

 be entirely defeated. Under these circumstances I felt obliged 

 to leave out z altogether. The comparative fixity thus given to 

 the arc a/378 on the variable ellipse I was aware of; and it is 

 this which Captain Clarke objects to, and because of it, con- 

 demns my method of correcting BessePs process, which he seems 

 to think needs no correction. But the omission of this quan- 

 tity z, which is sure to be very small, is not to be compared in 

 importance with the omission of t (which Bessel's method omits), 

 as t may be a comparatively large quantity. I do not think, 

 therefore, that on a careful reexamination of the subject Captain 

 Clarke will repeat his sentence, that I have " obscured " the 

 matter, and that, in the sense in which he means, my calculation 

 is " incorrect." I should rather turn the tables, and say that 

 " the elements of the figure of the earth deduced by " Bessel's 

 method, as he used it, " are, although they happen to be near 

 the truth, arbitrary results founded on an incorrect calculation," 



