383 



was strong, in this experiment was not a little curious, the perfect 

 image of the chess-board after bursting into view, gradually fading 

 altogether away, and then reviving, in less strung tints, in a series 

 of repetitions. 



Another curious, though anticipated result, the author also de- 

 scribes under this section, — the determination, by viewing the ocular 

 spectra, of portions of diagrams or elements in pictorial or typogra- 

 phical surfaces, which had not been noticed in the act of gazing. 

 Thus, particularly on viewing a line of printed figures at a particular 

 point, without noticing those on either side, a considerable series, 

 right and left, were so plainly depicted on the visual organ as to be 

 easily known ; whilst, in like manner, a point in a line of a printed 

 placard being gazed at, the lines above and below came into view on 

 closing the eyes, and could frequently be read. 



Of certain general facts elicited by this first series of investigations, 

 the author notices, that in viewing impressions on the retina with 

 closed eyes, all the pictures appear to occupy a position externally, 

 similar to the effect when the objects are directly seen ; that the 

 spectra derived from moderate or strong degrees of transmitted light 

 have prevalently the character of transparency, and those from very 

 low degrees, most ordinarily, of opacity; that although many of the 

 spectral phenomena the author had observed were well known to be 

 capable of elicitation in the ordinary form of the experiment with 

 the eyes open, yet the series of phenomena, as a whole, could not 

 be so elicited, nor vvas it possible by such form of experiment to 

 analyse and compare the phenomena whilst in progress of change, 

 which, in the form he had adopted, were usually exhibited as plainly 

 as if the spectra were the real and immediate eff^ects of ordinary 

 direct vision ; and that such is the precision and such the cer- 

 tainty with which the pictures are ordinarily developed, after duly 

 viewing any illuminated object, that the expected result, so far as the 

 eliciting of definite pictures is concerned, hardly ever fails. 



2. " On certain Properties of Square Numbers and other Quadratic 

 Forms, with a Table by which all the odd numbers up to 9211 may- 

 be resolved into not exceeding four square numbers." By Sir 

 Frederick Pollock, F.R.S. &c. Received Dec. 20, 1853. 



In examining the properties of the triangular numbers 0, 1, 3, 6, 

 10, &c., the author observed that every triangular number was com- 

 posed of four triangular numbers, viz. three times a triangular num- 

 ber plus the one above it or below it ; and he found that all the 

 natural numbers in the interval between ^nytico consecutive triangular 

 numbers might be composed of four triangular numbers having the 

 sum of their roots, or rather of the indices of their distances from 

 the first term of the series constant, viz. the sum of the indices of 

 the four triangular numbers which compose the first triangular 

 number of the tico. 



Not being at that time aware of any law by which the series that 

 fills up the intervals could be continued, he subsequently turned his 

 attention to the square numbers as apparently presenting a greater 



