84 Scientific Proceedings^ Royal Dublin Society. 



col. 2 contains the corresponding compound index numbers. The relation 

 between the index numbers is such that the sum of the simple index 

 numbers of any two units is equal to the compound index number of their 

 product. Table 3 is re.ally a re-arrangement of Table 2, the numbers 

 to 66 (representing 67 divisions on the index) being placed in col. 1, and 

 in col. 2, opposite to each number in col. 1 which is a compound index 

 number, is placed the corresponding simple product. 



Now, to take a very simple example, suppose the machine is supplied 

 with a formula-paper designed to cause it to evaluate as for given values 

 of a, b, 0, and d, in the equation ah + cd = x, and suppose we wish to 

 find the value of x in the particular ease where a = 9247, b = 8132, 

 c = 21893, and d = 828. 



The four given numbers are first transferred to the machine by the 

 key-board hereafter mentioned ; and the formula-paper causes them to be 

 inscribed in four sliuttles. As tlie shuttles of the inner and outer co-axial 

 shuttle-boxes are numbered consecutively, we may suppose the given values 

 of a and c to be inscribed in the first and second shuttles respectively of 

 the inner box, and of b and d in the first and second shuttles respectively of 

 the outer box ; but it is important to remember that it is a function of the 

 formula-paper to select the shuttles to receive the Variables, as well as the 

 shuttles to be operated on, so that (except under certain special circumstances, 

 which arise only in more complicated formulae) any given formula-paper 

 always selects the same shuttles in the same sequence and manner, whatever 

 be the values of the Variables. The magnitude of a Variable only effects 

 the type carried by its shuttle, and in no way influences the movements of 

 the sliuttle as a whole. 



The machine, guided by the formula-paper, now causes the shuttle- 

 boxes to rotate until tlie first shuttles of both inner and outer boxes come 

 opposite to a shuttle-race. The two shuttles are then drawn along the race 

 to a position near the index ; and certain slides are released, which move 

 forward until stopped by striking the type carried by the outer shuttle. 

 The slides in question will then have moved distances corresponding to 

 the simple index numbers of the corresponding digits of the Variables b. 

 In the particular case uuder consideration, the first four slides will tlieref ore 

 move 3, 0, 7, and 1 units respectively, the remainder of tlie slides indicating 

 zero by moving 50 units (see Table 1). Another slide moves in the opposite 

 direction until stopped by the first type of the inner shuttle, making a 

 movement proportional to the simple index number of the first digit of the 

 multiplier a — in this case 14. As the index is attached to the last- 

 mentioned slide, and partakes of its motion, the relative displacements of 



