4-26 



THE CIVIL ENGINEER AND ARCHITECTS JOURNAL. 



[December, 



^ow cD — cd=d D, tlien/cx c(l=F/x dViod 

 therefore Ff '■ fc : : cd : dD and by composition 

 Ff/.fo : fc : : cd^dD : (Z /) or simply 



Fc . fc : : cD : dD 



Now tlic aperture of the diapliragm which limits the field of view is un- 

 changeable in diameter in any part of its motion, and it is endent that wliile 

 it is at/it subtends or includes the angle id or t'ct', and while at /'the an- 

 gle »c« or s'cs', and that wliile at F the diaphracm ought to be enlarged to 

 the size tsFxl, to include the same angle that the same diaphragm rfr does 

 at/, and to read oti' a portion of the image of the staff, which would bo al- 

 ways proportional to the distance, for the angle remaining the same, the tan- 

 gent varies as the radius, or the subtense of the angle as the distance. 



Now because the number of divisions of any stalf s'lh' or I'DI' forming 

 the subtense of any fixed angle i'ct' is greater or smaller exactly in propor- 

 tion as the distance ^ ' ,.,i said staff from the angular point c is greater or 

 smaller, it is also e»k.rent that the diaphragm rfr or xfi will, while atf, show- 

 a larger angular proportion of the image of the staff than it does while at F, 

 and that the proportion will be as (Ft : sFs or as //■" to rf, but tF \ rf. : 

 Fc : fc and it has been shown that Fc '■ fc '. '. cD '. dD, therefore by in. 

 vertingWi) : cD.l : sFs : //7, and since //"j = sFs + 2.f/ then rf/) : cD: : 

 sFs '. sFs+2sl, that is to say, the distauce of the staff from the anterior 

 focus d of the object-glass is to the distance of the same staff from the object- 

 glass as the actual reading of the staff is to the reading augmented by a 

 cjuantity 2.«/'. 



'Sow dD : cD-dD : : sFs : sFs+2sl-sFs. 

 or simply dD '. ed : '. sFs '. 2st. 



That is to say, as the distance of the staff from the anterior focus of the 

 object-glass is to the principal focal distance, so is the actual reading of the 

 staff (xFs which is the only part of the image visible through the diaphragm) 

 to the quantity (2s/) which makes when added to sFs (the actual reading) an 

 augmented reading tFf, 



Now dD . sFs : : cd : 2sf,or — = — 

 • sFs 2st 



dD 



Again dD : sFs : : cD : tFt, or — = 



therefore — = = 



c_D 

 sFs^iFt' 

 cd_Dc 

 ~'27t~7Ft' 



Now cD and dD may be any quantities whatsoever, and as fFl and xFs 

 \ar\- resj)ectively as cD and dD, therefore IF/ and sFs are also variable quan- 

 tities, and may be of any dimensions ; but cd, the princi])al focal distance, is 

 an invariable quantity, therefore 2st is also an invariable quantity, for it varies 

 as erf varies, the variation of which is nothing; therefore, the variation of 

 2sl is also nothing, or 26/ is a constant quantity. 



To find practically this constant quantify 2s/ for any telescope and staff, 

 ■which 2»/, when added to the reading, makes an augmented reading always 

 propoitional to the distance of the oljjcct from the object-glass of the teles- 

 cope. First adjust the eye-piece to distinct vision of the cross lines upon the 

 diaphragm, and mark the sliding part of the eye-piece, so that it may after- 

 wards be kept to the same point ; then on some clear night, observe carefully 

 some star or planet, and wheu by moving the slide of the object-glass a little 

 to and fro, the sharpest and most cleaily defined image of this star or planet 

 has been obtained, mark carefully this point upon the slide of the object-glass 

 as the adjustment for the ])rincipal focus, and as the highest limit of a scale 

 to be afterwards graduated upon this slide ; the telescope being kept at this 

 adjustment, the distance between the cross lines on the diaphragm and the 

 object-glass will be the principal focal distance /c or cd. (Fig. No. 1.) 



Then the instrument being fixed in any convenient open place, measure 

 any distance cD, and observe the reading sFs, the image of part of the staff 

 seen over the diameter of the aperture of the diaphragm ; then as dD, that is 

 to say, the distance cD less the principal focal distance, is to the said reading, 

 so is the principal focal distance to the constant 2s/, which, when added to 

 the reading, makes an augmented reading, which is always proportional to 

 the actual distance of the object from the object-glass ; but as it would be 

 more convenient still to have a fixed quantity or constant to make the read- 

 ings when augmented by it always proportional to the distances from the 

 object to the centre of the instrument or station, and that such a quantitv 

 exists may be found out by accurate experiments, and may also be thus de- 

 monstrated. 



Let .V be the centre of the instrument or station (see Fig. No. 2) ; let c be 

 the object-glass ; let d be the anterior princijial focus of the object-glass ; let 

 D be any distance beyond this focus ; and let rDr rcjjresent the portion of 

 the staff visible over the diameter of the diaphragm of the telescope, or as it 

 is called simply the reading ; it has been shown that the reading rDr varies 

 as the distance dD varies ; from the centre of the object-glass c, draw ct, ct 

 parallel to dr, dr. and from the centre of the instrument A' draw Xw, Xw 

 parallel to dr, dr, and produce Dr, Dr, to /, / and w, w, then by similar 

 triingles. 



dU : rD: : cD : tD,novi2rD = rDr, and 2/Z) = /J[)/, and by transpos- 

 ing, &c., the above proportion (7iJ : cD : : 2rD : 2//) : : rDr : /fl/, but 

 it has been before shown that dn : cD '■ '. rDr, or the reading: to the read- 

 ing + a constant quantity; therefore //)/ = the reading + the constant, and it 



is equal to r/>r -r 2/r, therefore 2/r is the constant corresponding with 2/s in 

 Fig. 1. 



Fig. 2. 



Again, by the construction (similar triangles. Fig. 2) it is evident also 

 that <//) : XD : : rDr : uJHc, or is dl) :~Xd+dD '. '. rDr : rZ)r+ 2icr, 



and by subtraction dfl : Xd : : rDr : 2!tr, or -y^^-z — . now di) and riSr 

 ' rDr 2wr 



are variable quantities and vary as each other, but Xd is for the same instru- 

 ment an invariable quantity, and as 2wr varies as Xd varies, therefore 2wr is 

 also an invariable or a constant quantify, and it is the arooiuit which must be 

 added to the reading rDr to make a quantity ic/Jhi, which shall be propor- 

 tional to the distance of the object from the centre of the station, or by a 

 suitable construction of the diaphragm the quantity »c73h- shall be the actual 

 distance itself. 



From the inspection of Fig. No. 2, it is evident that if the parallel lines b 

 produced beyond wDw, and lines w'D'uj', w"Jy"w" &e. be drawn parallel to 

 it, that the readings r'D'r', r"D"r", &c., will increase or diminish as the dis- 

 tances dD', dD", &c., while the quantity 2«r will remain constant and always 

 equal to 2w'r', 2ui"r", &c., which by the construction of the figure, are all 

 equal froiu the point d where the readings of the staff begin, or are at Zero. 



Taking a practical example with the telescope of the 20 inch level before 

 mentioned, the distance cd measures 18"5 inches, or 2-33 links, the distance 

 Ac, or from the object-glass to the centre of the tripod, is 1-27 links, there- 

 fore the whole distance Xd is 3'6 links. Now, when the staff is held at 1000 

 Unks or 660 feet from the centre of the instrument, or 1000 — 3-6 = 996'4 

 Unks from the point d, the reading is 13-80 feet; now here f/n = 996-4, 

 A'/) = 1000, r/;r= 13-80 feet, and it is required to find w/)w and the con- 

 stant 2wr. 



Now dD : XD : : rDr : u-Dtc, or 



990-4 ; 1000 : : 13-80 : 13-S49 or 13-S.) nearly 

 and because wDw — rDr = 2wr, therefore 



13-849 — 13-80 = -049 = 2(i')-, or the constant which equals "05 nearly, and 

 the augmented reading is 13-85. 



Again, taking another example at the distance of 200 links, 2-"2 feet were 

 read off, now 200 -3-6 = 196-4 = <?/>, and 198-4 : 200: : 2-72 : 2-769 

 or 2-77 nearly and 2-769- 2-72 = -049 or -05 nearly, which is the constart 

 as in the former case, and the augmented reading ivDw in this case is 

 2-769 or 2-77 nearly. 



Therefore -05 is the constant number to he added to each reading to make 

 augmented readings proportional to the actual distances, when this instrument 

 and staff" graduated into feet and decimal parts, &e. are used. 



The scale for this instrument is, therefore, 1385 divisions, each Ihe-j-^tli 

 part of a foot, corresponding with 1000 links in distance, and therefore to 

 plot a section at the rate of 2000 links to the inch, the scale will have to be 

 a 27-7, instead of a 20 to the inch, when the divisions of the scale will plot 

 the readings in distance. 



The diameter of the diaphragm of this telescope is '39 inches, but by fixing 

 screws or points on the circumference of this diaphragm, the diameter of tiiis 

 aperture might be contracted, so that the distance between the points of flie 

 screws might not exceed the proportion of 1000 divisions of the stafl'to lOOO 

 Unks in distance ; but as the advantages peculiar to each (that is, whether it 

 is best to keep the instrument as it is, and read off proportional distances 

 only, or reduce the diameter and read oft" actual distances,) have been treated 

 of before in this paper, it is quite unnecessary here to go over the subject 

 again. 



A smaller level with an inverting 10 inch telescope, gives with the same 

 staff the constant quantify (from the centre of the station) -07 feet. 



The principal focal distance of this telescope is 10-5 inches, the diameter 

 of the aperture of the diaphragm -53 inches, the distance of the objecf-glass 

 from the centre of the tripod 6-4 inches;* by this instrument 1000 hnks in 



* Here it may be observed, that the distance between the object-glass a 

 the centre of the instrument OJght to be invariable, and that the increas 



