8 
MR, A. B. KEMPE OX THE THEORY OF MATHEMATICAL FORM. 
40. A single heap may be graphically represented by a number of small separated 
circles each containing the same letter. (Fig. 4.) 
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Fig. 4. 
41. Differences arising from the positions of the graphical units on the paper on 
which they are drawn are always to be disregarded. Where positions are considered, 
they must be represented by separate graphical units. 
42. Where there is occasion to employ letters outside and adjacent to the graphical 
units for purposes of reference, differences arising from the use of such letters are also 
to be ignored. (Sec. 173.) 
43. In every system of n units other then a discrete or single heap, some s-ads are 
distinguished from each other and some are not, for all or some values of s from 1 to n. 
Thus every form which a system of n units can assume may be regarded as either that 
of a discrete or single heap or as intermediate between the two. 
44. There are systems of n units of other forms than those of discrete and single 
heaps which can be represented by means of graphical units alone. These consist of 
one or more detached independent single heaps. The term “ independent ” will be 
fully explained presently (secs. 117, 118) ; for the present it will be sufficient to state 
that two independent systems are such as can be graphically represented on separate 
sheets of paper, without the employment of symbols on either sheet relating to those 
on the other. 
45. A system of s independent detached heaps will be termed an s -tuple heap. 
Fig. 5 represents a treble heap of seven units. 
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Fig. 5. 
46. A discrete heap of n units is an w-tuple heap of n units. 
47. The number of different forms which a heap of n units can assume is the 
same as that of the partitions of n. 
48. It will in some cases be convenient, instead of employing circular graphical 
units distinguished by internal letters, to use coloured graphical units, or black spots 
of different sizes, or graphical units of various shapes. 
49. Where graphical units are employed alone, like units will represent undis¬ 
tinguished units, but in the case of forms other than heaps it will be necessary 
to emjrloy means to distinguish pairs, &c., and these may cause like graphical units 
to be distinguished from each other. No general inference must therefore be drawn 
from the use of like and unlike graphical units other than that unlike units represent 
distinguished units. 
