Hello Frances,
From other army records I have for family members their age, physical condition, address and next of kin is recorded. However these records only pertain to WW1. When I discovered Alexander junior joined the RIR I was hoping for a next of kin or address so I could establish his whereabouts.

The fact he drowned in Derbyhaven in the IOM leads me to believe this was his home base. This is a response from the geneology site of the Belfast Forum.
No sign of any service record for him, and I'm not surprised. Service records at that time were mostly only retained for men who were eligible for a pension, and he was only in the Army for less than a year.

At that time regiments or corps each issued new recruits to that unit with his regimental number, which at that time wasn't unique to an individual, because many different regiments and corps were using the same number blocks.

In the Royal Irish Rifles...
9239 enlisted 2nd January� 1909
9284 ?
9377 enlisted 24th January 1910

9284 died 18th September 1909 So he was probably only enlisted for two or three months, and up to a maximum of 9.

Between 1906 and 1911 the 1st Battalion was in the Far East...2nd battalion was in Aldershot

He was buried in Malew on 20th September 1909, near where he died. An image is available on the LDS Family search where I discovered additional writing next to the entry.

Jane Miller his mother is also buried on the island (30 October 1890). Her last address is 4 Fairy Grounds which I believe is in Douglas. I dont believe the connection with his mother was determined and they are buried in seperate places,