The Anglo-Boer website http://www.angloboerwar.com/
list him as:
Robertson; John Ainslie Sommerville; 15730; Private; No known Company. Served in 20th Btn IY
Source: QSA Medal Rolls.
However, you should be aware that there is another Robertson I.Y. soldier with the same number (confirmed in the Nominal Roll of the I.Y. on Kevin Asplin’s “The Asplin Military History Resources”) that may be a pointer that either one of the men’s numbers is wrong or they are the same man, who re-enlisted after his initial year of service, albeit with a different first name; he was:
Robertson, James S; 15730; Sergeant; Pay Office Staff I.Y.; Source: QSA Medal Rolls
The breakdown of the 20th Battalion I.Y. is given as:
20th Battalion (Rough Riders) raised 17 Mar. 1900 in London; disbanded July 1901; perpetuated 23 July 1901 by City of London Imperial Yeomanry (Rough Riders).
72nd (Rough Riders) Company, raised 1900; disbanded 1900
76th (Rough Riders) Company, raised 1900; transferred 1902 to 22nd Battalion
78th (Rough Riders) Company, raised 1900; transferred 1902 to 22nd Battalion
79th (Rough Riders) Company, raised 1900; disbanded 1900
You’ll find an introduction to the I.Y., it’s history etc at http://angloboerwar.com/name-search/1946
Attestation & Discharge papers for the Imperial Yeomanry are not available online anywhere, but are held at the National Archives, Kew, filed by regimental number in document series WO 128 War Office: Imperial Yeomanry, Soldiers' Documents, South African War; 1899-1902.
The record for regimental number 15730 is in piece number:
WO 128/48 War Office: Imperial Yeomanry, Soldiers' Documents, South African War; Regimental Numbers; 15,711 - 15,910; dates 1899-1902.
These attestation & discharge documents don’t normally contain family information, just a soldier’s place of birth, age, and occupation and details of his service.
If you cannot get to Kew your only viable cost effective option to obtain a copy would be to use a professional researcher. If you wish to pursue that option then from the NA’s home page at http://www.nationalarchives.gov.uk/
select Records > (scroll down) Paying for Research (note the NA’s charges!) > Independent Researchers > select Army etc.
You will find a list of researchers, several with web sites and emails, all charges are plus copies but beware “plus expenses”, the rates for reputable researchers should include their “expenses”, average cost should be £20-£30 /hr.
For a quick source use your browser to search for “military research on line” and one of the first listed should a company, you will also find via the longer route, that offers fixed price packages for some records and a reasonable hourly rate for longer searches.
There is an online database of Boer War soldiers at the pay per view site Findmypast that gives some additional information that appears to confirm that John Ainslie Sommerville Robertson did become a sergeant and is listed as being in the 78th Co, 20th Battalion I.Y. References quoted Records of the Rough Riders: McKenzie Rew 277
The National Archives WO128. Imperial Yeomanry, Soldiers' Documents, South African War.
The similar record for James S Robertson also records him as sergeant “Pay Office Imperial Yeomanry”, so appears to add more evidence the two men are one and the same.