OiX TLAAIMETERS. 5L9 



(p. 502) the area generated equals (T) — (Q), where (T) is the area of the 

 given curve and (Q) the area of the closed Q-curve, or equal to twice the 

 sector AOB, i.e., equal to I. a, where a denotes the arc OB. If we now 

 could vary the proceeding so that (Q):=0 we should have (T)^la, and 

 need only measure the arc a to get the area of the given curve. 



To do this draw from A any straiglit line AX, place the planimeter in 

 its original position OA, and move T from A along AX. The point Q now 

 describes a tractrix OF with AX as asymptote. On doing the same from 

 the end-position BA we get a second tractrix, BH, also with AX as 

 asymptote. 



If we now take on AX a suitable point R and draw with this point as 

 centre and TQ as radius an arc ED between these two tractrices, we can 

 consider the curve DOSBED as the Q-curve. We need only start at the 

 position DR for QT, move T to A, then round the curve and back to R, 

 turning QT at last about R till Q comes back to D. 



If R has been well selected the arc ED will cut the curve OSB, and 

 therefore the area of the Q-cur^e will be partly positive, partly negative. 

 If, in tig. 16, the point R is moved higher up, the positive part will increase ; 

 on moving it lower down, the negative. Hence there must exist a definite 

 position for R such that the area (Q) vanishes. 



This shows the possibility of using the simple ' Hatchet ' as a plani- 

 meter ; but it is not yet a practical instrument. The above shows only : — 



If a point A, a line AX, and an initial position OA are taken 

 arbitrarily, then a point R exists on AX such that on starting with T at 

 R and Q at D we get the area reduced to twice the area of a sector with 

 radius QT. 



We have here at our disposal first the point A on the curve ; secondly, 

 the direction of the line AX ; thirdly, the initial direction of QT, for it 

 comes to the same whether we take OA or DR as given. These three 

 being fixed, the above proves the existence of a point R. The change of 

 one of these three quantities alters the position of R. 



We must, in order to get a practically useful rule for determining 

 R, restrict the superabundance of choice which the above theory leaves 

 us. A perfectly satisfactory rule has not yet been found. The only 

 generally usable one is that given by the inventor. 



As the Hatchet Planimeter has during the last few years excited 

 some interest both in England and abroad, and as I have heard its 

 invention atti-ibuted to various men, the following historical facts may be 

 mentioned. 



I became acquainted with it early in 1893 through Professor Green- 

 hill, and ordered one from the maker, Herr Cornelius Knudsen, in Copen- 

 hagen. With it I received a pamphlet in French dated 1887. It contains 

 an analytical theory without mentioning Captain Prytz. After my showing 

 the instrument at the Physical Society and mentioning that a complete 

 theory did not yet exist, thei-e appeared in ' Engineering ' a paper by 

 Macfarlane Gi-ay. In consequence of this Captain Prytz contributed his 

 investigations to the same journal (where his name is changed to Pryty). 

 It is practically an English translation of the pamphlet mentioned. I 

 have since seen a new pamphlet with the name of Captain Prytz on it. 

 In England the instrument seems, however, to have first become known 

 through Mr. Druitt Halpin, who mentioned it to Professor Unwin, 

 Professor Goodman, and other engineers about 1889. Professor Goodman 



