38 REPORT 1841. 



of an elliptic edge, and e be taken to represent the ratio of the eccentricity of this 

 ellipse to its axis major, and if its semi-axis minor be taken = 2 «■ g, then will the 

 number registered be represented by the formula 



N= M ^^ . 

 y Vl — e^sin^tf 



So that the machine may be made with this rail to calculate an elliptic function of 

 the first order of any amplitude 6 whose modulus is e. 



4th. That if the form of the rail were that of a logarithmic spiral whose pole co- 

 incided with the centre of the plate, and y representing the constant angle of the 

 spiral and a the radius vector from which was measured, if these quantities were 



so taken that ?i?fL2'= i and £*^°*^= 10, then would the numbers registered be 



2 7rq 

 represented by the formula j^ i j __ jq^. 



The arc & is therefore the logarithm of the number N + 1 greater than that regis- 

 tered by unity ; and if, connected with the endless screw which gives motion to the 

 plate, there be a train of wheels registering the space described by a point in the 

 plate at distance unit)' from its centre to four places of decimals, and beginning from 

 that position of the plate in which the given radius vector a coincides with the dia- 

 meter described by the point of contact of the plate and wheel, then when the regis- 

 ter which gives the revolutions of the integrating wheel is made to show one less 

 any given number N, that which gives the revolution of the plate will show a number 

 which is the common logarithm of N ; so that under this form the machine may be 

 made to replace a table of common logarithms. The surface of the plate might in 

 this particular case be advantageously replaced by the surface of a cone, and the 

 spiral would then become identical with the natural spiral traced out by the convo- 

 lution of a large class of turbinated shells*. The preceding examples of the calcu- 

 lations which may be performed by the machine are examples whose solutions lie 

 within the ordinary resources of analysis, in respect to which the instrument could 

 only serve to replace, perhaps conveniently, but (except it were constructed on a large 

 scale) less accurately, well known processes of analytical calculation or tables already 

 in use. 



Professor Moseley next proceeded to show that there is a large class of func- 

 tions whose analytical integration lies beyond the existing resources of mathema- 

 tical science, the mechanical integration of which this machine would nevertheless 

 effect. He then stated, that among the mechanical difficulties which would present 

 themselves to the construction of such a machine were the following. Great accu- 

 racy would be required in giving their true geometrical forms to the curves fixed 

 upon the revolving plate. It is unquestionable that a certain amount of error must 

 always remain due to inaccuracies of workmanship in the forms of these curves ; 

 nevertheless to those who are acquainted with the wonderful education of the sense 

 of touch and skill of hand acquired in the minute workmanship of some processes of 

 the arts, (the engraving of tj'pe-founders' dies for instance, or the drawing of micro- 

 metrical lines on glass,) it will not appear impossible to reach, with the requisite care 

 and patience, a very considerable degree of precision in this respect ; and let it be 

 remembered, that one type or model being thus attained, it may be reproduced infi- 

 nitely, and with perfect accuracy, by casting in type-founders' metal, or perhaps by 

 turning, or by the electrotj'pe. Again, the difficulty of constructing these curv'es 

 would, no doubt, in some degree be enhanced by the fact, that the guiding curve 

 must be different from what it would be if the integrating wheel at its point of con- 

 tact with the plate were allowed to be in contact with the face of the rail. This difficulty 

 (not in itself considerable) may, however, wholly be removed. We have only to take 

 the curves from the plate, and to place them on another circular plate parallel to and 

 concentric with it, of precisely the same dimensions, but so far above it as to clear 



* This property of turbinated shells was published by Prof. Moseley in a paper on the Geo- 

 metrical Forms of Turbinated and Discord Shells in the Phil. Trans, for 1838. It lias since been 

 confirmed by the admeasurements of Prof. Naumann, of Friburg, published in the Journal 

 of Poggendorff for 1840. 



